Search: a277996 -id:a277996
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A317882
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Number of free pure achiral multifunctions (with empty expressions allowed) with one atom and n positions.
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+10
6
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1, 1, 2, 5, 12, 31, 79, 211, 564, 1543, 4259, 11899, 33526, 95272, 272544, 784598, 2270888, 6604900, 19293793, 56581857, 166523462, 491674696, 1455996925, 4323328548, 12869353254, 38396655023, 114803257039, 343932660450, 1032266513328, 3103532577722
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OFFSET
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1,3
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COMMENTS
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A free pure achiral multifunction (with empty expressions allowed) (AME) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty expression of the form h[g, ..., g] where h and g are AMEs. The number of positions in an AME is the number of brackets [...] plus the number of o's.
Also the number of achiral Mathematica expressions with one atom and n positions.
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LINKS
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FORMULA
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a(1) = 1; a(n > 1) = a(n - 1) + Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1)} a(d).
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EXAMPLE
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The a(5) = 12 AMEs:
o[o[o]]
o[o][o]
o[o[][]]
o[o,o,o]
o[][o[]]
o[][o,o]
o[][][o]
o[o[]][]
o[o,o][]
o[][o][]
o[o][][]
o[][][][]
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MATHEMATICA
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a[n_]:=If[n==1, 1, Sum[a[k]*If[k==n-1, 1, Sum[a[d], {d, Divisors[n-k-1]}]], {k, n-1}]];
Array[a, 12]
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PROG
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(PARI) seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*(1 + sum(k=1, n-2, subst(p + O(x^(n\k+1)), x, x^k)) ) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018
(PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=v[n-1] + sum(i=1, n-2, v[i]*sumdiv(n-i-1, d, v[d]))); v} \\ Andrew Howroyd, Aug 19 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A317883
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Number of free pure achiral multifunctions with one atom and n positions.
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+10
6
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1, 0, 1, 1, 3, 4, 10, 17, 37, 70, 150, 299, 634, 1311, 2786, 5879, 12584, 26904, 58005, 125242, 271819, 591297, 1290976, 2825170, 6199964, 13635749, 30057649, 66386206, 146903289, 325637240, 723024160, 1607805207, 3580476340, 7984266625, 17827226469
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OFFSET
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1,5
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COMMENTS
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A free pure achiral multifunction (PAM) is either (case 1) the leaf symbol "o", or (case 2) a nonempty expression of the form h[g, ..., g] where h and g are PAMs. The number of positions in a PAM is the number of brackets [...] plus the number of o's.
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LINKS
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FORMULA
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a(1) = 1; a(n > 1) = Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1)} a(d).
G.f. A(x) satisfies: A(x) = x * (1 + A(x) * Sum_{k>=1} A(x^k)). - Ilya Gutkovskiy, May 03 2019
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EXAMPLE
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The a(7) = 10 PAMs:
o[o[o[o]]]
o[o[o][o]]
o[o][o[o]]
o[o[o]][o]
o[o][o][o]
o[o[o,o,o]]
o[o][o,o,o]
o[o,o][o,o]
o[o,o,o][o]
o[o,o,o,o,o]
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MATHEMATICA
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a[n_]:=If[n==1, 1, Sum[a[k]*Sum[a[d], {d, Divisors[n-k-1]}], {k, n-2}]];
Array[a, 12]
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PROG
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(PARI) seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*sum(k=1, n-2, subst(p + O(x^(n\k+1)), x, x^k) ) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018
(PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=sum(i=1, n-2, v[i]*sumdiv(n-i-1, d, v[d]))); v} \\ Andrew Howroyd, Aug 19 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A317884
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Number of series-reduced achiral free pure multifunctions (with empty expressions allowed) with one atom and n positions.
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+10
6
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1, 1, 1, 2, 4, 8, 14, 26, 47, 87, 160, 295, 540, 997, 1832, 3369, 6197, 11406, 20975, 38594, 70991, 130610, 240275, 442043, 813184, 1496053, 2752251, 5063319, 9314879, 17136632, 31526032, 57998423, 106699160, 196294065, 361120800, 664352454, 1222204958
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OFFSET
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1,4
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COMMENTS
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A series-reduced achiral expression (SRAE) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty but not unitary expression of the form h[g, ..., g], where h and g are SRAEs. The number of positions in an SRAE is the number of brackets [...] plus the number of o's.
Also the number of series-reduced achiral Mathematica expressions with one atom and n positions.
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LINKS
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FORMULA
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a(1) = 1; a(n > 1) = a(n-1) + Sum_{0 < k < n-1} a(k) * Sum_{d|(n-k-1), d < n-k-1} a(d).
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EXAMPLE
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The a(6) = 8 SRAEs:
o[o,o,o,o]
o[o[],o[]]
o[][o,o,o]
o[][][o,o]
o[o,o,o][]
o[][o,o][]
o[o,o][][]
o[][][][][]
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MAPLE
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a:= proc(n) option remember; `if`(n=1, 1, a(n-1)+add(a(j)*add(
a(d), d=numtheory[divisors](n-j-1) minus {n-j-1}), j=1..n-1))
end:
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MATHEMATICA
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allAchExprSR[n_] := If[n == 1, {"o"}, Join @@ Cases[Table[PR[k, n - k - 1], {k, n - 1}], PR[h_, g_] :> Join @@ Table[Apply @@@ Tuples[{allAchExprSR[h], Select[Tuples[allAchExprSR /@ p], SameQ @@ # &]}], {p, If[g == 0, {{}}, Join @@ Permutations /@ Rest[IntegerPartitions[g]]]}]]]; Table[Length[allAchExprSR[n]], {n, 12}]
(* Second program: *)
a[n_] := a[n] = If[n == 1, 1, a[n-1] + Sum[a[j]*DivisorSum[
n-j-1, If[# < n-j-1, a[#], 0]&], {j, 1, n-2}]];
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PROG
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(PARI) seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*(1 + sum(k=2, n-2, subst(p + O(x^(n\k+1)), x, x^k)) ) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018
(PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=v[n-1] + sum(i=1, n-2, v[i]*sumdiv(n-i-1, d, if(d<n-i-1, v[d], 0)))); v} \\ Andrew Howroyd, Aug 19 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A317853
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a(1) = 1; a(n > 1) = Sum_{0 < k < n} (-1)^(n - k - 1) a(n - k) Sum_{d|k} a(d).
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+10
5
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1, 1, 1, 1, 2, 2, 5, 6, 11, 14, 23, 26, 51, 70, 114, 147, 237, 314, 516, 715, 1118, 1549, 2353, 3252, 5011, 7235, 10724, 15142, 22504, 32506, 47770, 69173, 100980, 146657, 212504, 308563, 448256, 658037, 946166, 1373739, 1988283, 2919185, 4197886, 6118850
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OFFSET
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1,5
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LINKS
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MATHEMATICA
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a[n_]:=a[n]=If[n==1, 1, Sum[(-1)^(n-k-1)*a[n-k]*Sum[a[d], {d, Divisors[k]}], {k, n-1}]];
Array[a, 50]
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CROSSREFS
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Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A277996, A280000, A317875.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A318149
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e-numbers of free pure symmetric multifunctions with one atom.
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+10
5
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1, 4, 16, 36, 128, 256, 441, 1296, 2025, 16384, 21025, 65536, 77841, 194481, 220900, 279936, 1679616, 1803649, 4100625, 4338889, 268435456, 273571600, 442050625, 449482401, 1801088541, 4294967296, 4334247225, 6059221281
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OFFSET
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1,2
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COMMENTS
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If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique orderless expression e(n) (as can be represented in functional programming languages such as Mathematica) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1). The sequence consists of all numbers n such that e(n) contains no empty subexpressions f[].
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LINKS
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EXAMPLE
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The sequence of free pure symmetric multifunctions with one atom "o", together with their e-numbers begins:
1: o
4: o[o]
16: o[o,o]
36: o[o][o]
128: o[o[o]]
256: o[o,o,o]
441: o[o,o][o]
1296: o[o][o,o]
2025: o[o][o][o]
16384: o[o,o[o]]
21025: o[o[o]][o]
65536: o[o,o,o,o]
77841: o[o,o,o][o]
194481: o[o,o][o,o]
220900: o[o,o][o][o]
279936: o[o][o[o]]
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MATHEMATICA
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nn=1000;
radQ[n_]:=If[n==1, False, GCD@@FactorInteger[n][[All, 2]]==1];
rad[n_]:=rad[n]=If[n==0, 1, NestWhile[#+1&, rad[n-1]+1, Not[radQ[#]]&]];
Clear[radPi]; Set@@@Array[radPi[rad[#]]==#&, nn];
exp[n_]:=If[n==1, "o", With[{g=GCD@@FactorInteger[n][[All, 2]]}, Apply[exp[radPi[Power[n, 1/g]]], exp/@Flatten[Cases[FactorInteger[g], {p_?PrimeQ, k_}:>ConstantArray[PrimePi[p], k]]]]]];
Select[Range[nn], FreeQ[exp[#], _[]]&]
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PROG
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(Python) See Neder link.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A318150
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e-numbers of free pure functions with one atom.
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+10
5
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1, 4, 36, 128, 2025, 21025, 279936, 4338889, 449482401, 78701569444, 373669453125, 18845583322500, 1347646586640625, 202054211912421649, 6193981883008128893161, 139629322539586311507076, 170147232533595290155627, 355156175404848064835984400
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graph;
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listen;
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OFFSET
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1,2
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COMMENTS
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If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique orderless expression e(n) (as can be represented in functional programming languages such as Mathematica) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1). This sequence consists of all numbers n such that e(n) contains no non-unitary subexpressions f[x_1, ..., x_k] where k != 1.
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LINKS
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FORMULA
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a(1) = 1, and if a and b are in this sequence then so is rad(a)^prime(b). - Charlie Neder, Feb 23 2019
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EXAMPLE
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The sequence of all free pure functions with one atom together with their e-numbers begins:
1: o
4: o[o]
36: o[o][o]
128: o[o[o]]
2025: o[o][o][o]
21025: o[o[o]][o]
279936: o[o][o[o]]
4338889: o[o][o][o][o]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A318152
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e-numbers of unlabeled rooted trees. A number n is in the sequence iff n = 2^(prime(y_1) * ... * prime(y_k)) for some k > 0 and y_1, ..., y_k already in the sequence.
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+10
4
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1, 4, 16, 128, 256, 16384, 65536, 268435456, 4294967296, 562949953421312, 9007199254740992, 72057594037927936, 18446744073709551616, 316912650057057350374175801344, 81129638414606681695789005144064, 5192296858534827628530496329220096
(list;
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OFFSET
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1,2
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COMMENTS
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If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique orderless expression e(n) (as can be represented in functional programming languages such as Mathematica) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1). The sequence consists of all numbers n such that e(n) contains no empty subexpressions f[] or subexpressions in heads f[x_1, ..., x_k][y_1, ..., y_k] where k,j >= 0.
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LINKS
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EXAMPLE
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The sequence contains 16384 = 2^14 = 2^(prime(1) * prime(4)) because 1 and 4 both already belong to the sequence.
The sequence of unlabeled rooted trees with e-numbers in the sequence begins:
1: o
4: (o)
16: (oo)
128: ((o))
256: (ooo)
16384: (o(o))
65536: (oooo)
. (oo(o))
. (ooooo)
. ((o)(o))
((oo))
(ooo(o))
(oooooo)
(o(o)(o))
(o(oo))
(oooo(o))
(ooooooo)
(oo(o)(o))
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MATHEMATICA
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baQ[n_]:=Or[n==1, MatchQ[FactorInteger[n], {{2, _?(And@@Cases[FactorInteger[#], {p_, k_}:>baQ[PrimePi[p]]]&)}}]];
Select[2^Range[0, 50], baQ]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A318153
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Number of antichain covers of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.
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+10
4
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1, 2, 3, 2, 4, 3, 5, 3, 3, 4, 6, 4, 4, 5, 7, 2, 5, 5, 6, 8, 3, 6, 6, 7, 4, 9, 5, 4, 7, 7, 8, 4, 5, 10, 6, 3, 5, 8, 8, 9, 5, 6, 11, 7, 4, 6, 9, 9, 5, 10, 6, 7, 12, 8, 5, 7, 10, 10, 6, 11, 7, 8, 13, 3, 9, 6, 8, 11, 11, 7, 12, 8, 9, 14, 4, 10, 7, 9, 12, 12, 3, 8
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OFFSET
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1,2
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COMMENTS
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If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction (with empty expressions allowed) e(n) (as can be represented in functional programming languages such as Mathematica) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1). The a(n) is the number of ways to partition e(n) into disjoint subexpressions such that all leaves are covered by exactly one of them.
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LINKS
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FORMULA
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If n = rad(x)^(Product_i prime(y_i)^z_i) where rad = A007916 then a(n) = 1 + a(x) * Product_i a(y_i)^z_i.
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EXAMPLE
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441 is the e-number of o[o,o][o] which has antichain covers {o[o,o][o]}, {o[o,o], o}, {o, o, o, o}}, corresponding to the leaf-colorings 1[1,1][1], 1[1,1][2], 1[2,3][4], so a(441) = 3.
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MATHEMATICA
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nn=20000;
radQ[n_]:=If[n==1, False, GCD@@FactorInteger[n][[All, 2]]==1];
rad[n_]:=rad[n]=If[n==0, 1, NestWhile[#+1&, rad[n-1]+1, Not[radQ[#]]&]];
Clear[radPi]; Set@@@Array[radPi[rad[#]]==#&, nn];
a[n_]:=If[n==1, 1, With[{g=GCD@@FactorInteger[n][[All, 2]]}, 1+a[radPi[n^(1/g)]]*Product[a[PrimePi[pr[[1]]]]^pr[[2]], {pr, If[g==1, {}, FactorInteger[g]]}]]];
Array[a, 100]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A317676
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Triangle whose n-th row lists in order all e-numbers of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.
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+10
3
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1, 2, 3, 4, 5, 6, 8, 9, 16, 7, 10, 12, 13, 21, 25, 27, 32, 36, 64, 81, 128, 256, 11, 14, 17, 18, 28, 33, 35, 41, 45, 49, 75, 93, 100, 125, 144, 145, 169, 216, 243, 279, 441, 512, 625, 729, 1024, 1296, 2048, 2187, 4096, 6561, 8192, 16384, 65536, 524288, 8388608, 9007199254740992
(list;
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OFFSET
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1,2
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COMMENTS
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Given a positive integer n we construct a unique free pure symmetric multifunction e(n) by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)].
Every free pure symmetric multifunction (with empty expressions allowed) f with one atom and n positions has a unique e-number n such that e(n) = f, and vice versa, so this sequence is a permutation of the positive integers.
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LINKS
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EXAMPLE
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Triangle begins:
1
2
3 4
5 6 8 9 16
7 10 12 13 21 25 27 32 36 64 81 128 256
Corresponding triangle of free pure symmetric multifunctions (with empty expressions allowed) begins:
o,
o[],
o[][], o[o],
o[][][], o[o][], o[o[]], o[][o], o[o,o].
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MATHEMATICA
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maxUsing[n_]:=If[n==1, {"o"}, Join@@Cases[Table[PR[k, n-k-1], {k, n-1}], PR[h_, g_]:>Join@@Table[Apply@@@Tuples[{maxUsing[h], Union[Sort/@Tuples[maxUsing/@p]]}], {p, IntegerPartitions[g]}]]];
radQ[n_]:=And[n>1, GCD@@FactorInteger[n][[All, 2]]==1];
Clear[rad]; rad[n_]:=rad[n]=If[n==0, 1, NestWhile[#+1&, rad[n-1]+1, Not[radQ[#]]&]];
ungo[x_?AtomQ]:=1; ungo[h_[g___]]:=rad[ungo[h]]^(Times@@Prime/@ungo/@{g});
Table[Sort[ungo/@maxUsing[n]], {n, 5}]
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CROSSREFS
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Cf. A007916, A052409, A052410, A052893, A053492, A215366, A279944, A280000, A299759, A317658, A317659, A317677.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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A304485
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Regular triangle where T(n,k) is the number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions and k leaves.
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+10
2
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1, 1, 0, 1, 2, 0, 1, 6, 4, 0, 1, 12, 23, 7, 0, 1, 20, 81, 73, 12, 0, 1, 30, 209, 407, 206, 19, 0, 1, 42, 451, 1566, 1751, 534, 30, 0, 1, 56, 858, 4711, 9593, 6695, 1299, 45, 0, 1, 72, 1494, 11951, 39255, 51111, 23530, 3004, 67, 0, 1, 90, 2430, 26752, 130220, 278570, 245319, 77205, 6664, 97, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,5
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COMMENTS
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A free pure symmetric multifunction (with empty expressions allowed) f in EOME is either (case 1) a positive integer, or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where k >= 0, h is in EOME, each of the g_i for i = 1, ..., k is in EOME, and for i < j we have g_i <= g_j under a canonical total ordering of EOME, such as the Mathematica ordering of expressions.
T(n,k) is also the number of inequivalent colorings of orderless Mathematica expressions with n positions and k leaves.
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LINKS
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EXAMPLE
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Inequivalent representatives of the T(5,3) = 23 Mathematica expressions:
1[][1,1] 1[1,1][] 1[1][1] 1[1[1]] 1[1,1[]]
1[][1,2] 1[1,2][] 1[1][2] 1[1[2]] 1[1,2[]]
1[][2,2] 1[2,2][] 1[2][1] 1[2[1]] 1[2,1[]]
1[][2,3] 1[2,3][] 1[2][2] 1[2[2]] 1[2,2[]]
1[2][3] 1[2[3]] 1[2,3[]]
Triangle begins:
1
1 0
1 2 0
1 6 4 0
1 12 23 7 0
1 20 81 73 12 0
1 30 209 407 206 19 0
1 42 451 1566 1751 534 30 0
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PROG
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(PARI) \\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(p=O(x)); for(n=1, n, p = x*sv(1) + x*p*sExp(p)); p}
T(n)={my(v=Vec(InequivalentColoringsSeq(sFuncSubst(cycleIndexSeries(n), i->sv(i)*y^i)))); vector(n, n, Vecrev(v[n]/y, n))}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 01 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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