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Search: a317884 -id:a317884
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A317875
Number of achiral free pure multifunctions with n unlabeled leaves.
+10
13
1, 1, 3, 9, 30, 102, 369, 1362, 5181, 20064, 79035, 315366, 1272789, 5185080, 21296196, 88083993, 366584253, 1533953100, 6449904138, 27238006971, 115475933202, 491293053093, 2096930378415, 8976370298886, 38528771056425, 165784567505325
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
An achiral free pure multifunction 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 both achiral free pure multifunctions.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
FORMULA
a(1) = 1; a(n > 1) = Sum_{0 < k < n} a(n - k) * Sum_{d|k} a(d).
From Ilya Gutkovskiy, Apr 30 2019: (Start)
G.f. A(x) satisfies: A(x) = x + A(x) * Sum_{k>=1} A(x^k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x + (Sum_{n>=1} a(n)*x^n) * (Sum_{n>=1} a(n)*x^n/(1 - x^n)). (End)
EXAMPLE
The first 4 terms count the following multifunctions.
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[o]][o].
MATHEMATICA
a[n_]:=If[n==1, 1, Sum[a[n-k]*Sum[a[d], {d, Divisors[k]}], {k, n-1}]];
Array[a, 12]
PROG
(PARI) seq(n)={my(p=O(x)); for(n=1, n, p = x + p*(sum(k=1, n-1, 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-1, v[i]*sumdiv(n-i, d, v[d]))); v} \\ Andrew Howroyd, Aug 19 2018
CROSSREFS
Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A277996, A280000, A317853.
Cf. A317876, A317877, A317878, A317879, A317880, A317881.
Cf. A317882, A317883, A317884, A317885.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 09 2018
STATUS
approved
A317882
Number of free pure achiral multifunctions (with empty expressions allowed) with one atom and n positions.
+10
6
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
FORMULA
a(1) = 1; a(n > 1) = a(n - 1) + Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1)} a(d).
EXAMPLE
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[][][][]
MATHEMATICA
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]
PROG
(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
CROSSREFS
Cf. A001003, A002033, A003238, A052893, A053492, A214577, A277996, A317853, A317875.
Cf. A317883, A317884, A317885.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 09 2018
EXTENSIONS
Terms a(13) and beyond from Andrew Howroyd, Aug 19 2018
STATUS
approved
A317883
Number of free pure achiral multifunctions with one atom and n positions.
+10
6
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
FORMULA
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
EXAMPLE
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]
MATHEMATICA
a[n_]:=If[n==1, 1, Sum[a[k]*Sum[a[d], {d, Divisors[n-k-1]}], {k, n-2}]];
Array[a, 12]
PROG
(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
CROSSREFS
Cf. A002033, A003238, A052893, A053492, A214577, A277996, A280000, A317853, A317875.
Cf. A317882, A317884, A317885.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 09 2018
EXTENSIONS
Terms a(13) and beyond from Andrew Howroyd, Aug 19 2018
STATUS
approved
A317885
Number of series-reduced free pure achiral multifunctions with one atom and n positions.
+10
6
1, 0, 0, 1, 1, 1, 2, 3, 4, 7, 9, 14, 21, 32, 45, 69, 103, 153, 224, 338, 500, 746, 1107, 1645, 2447, 3652, 5413, 8052, 11993, 17834, 26500, 39447, 58655, 87240, 129772, 193001, 287034, 427014, 635048, 944501, 1404910, 2089633, 3107864, 4622670, 6875533
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,7
COMMENTS
A series-reduced free pure achiral multifunction (SRAM) is either (case 1) the leaf symbol "o", or (case 2) a nonempty and non-unitary expression of the form h[g, ..., g] where h and g are SRAMs. The number of positions in a SRAM is the number of brackets [...] plus the number of o's.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..500
FORMULA
a(1) = 1; a(n > 1) = Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1), d < n - k - 1} a(d).
EXAMPLE
The a(10) = 7 SRAMs:
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]
o[o,o,o,o,o,o,o,o]
MATHEMATICA
a[n_]:=If[n==1, 1, Sum[a[k]*Sum[a[d], {d, Most[Divisors[n-k-1]]}], {k, n-2}]];
Array[a, 12]
PROG
(PARI) seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*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]=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
CROSSREFS
Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A317853, A317875.
Cf. A317882, A317883, A317884.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 09 2018
EXTENSIONS
Terms a(17) and beyond from Andrew Howroyd, Aug 19 2018
STATUS
approved
A317853
a(1) = 1; a(n > 1) = Sum_{0 < k < n} (-1)^(n - k - 1) a(n - k) Sum_{d|k} a(d).
+10
5
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
LINKS
Table of n, a(n) for n=1..44.
MATHEMATICA
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]
CROSSREFS
Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A277996, A280000, A317875.
Cf. A317876, A317877, A317878, A317879, A317880, A317881.
Cf. A317882, A317883, A317884, A317885.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 09 2018
STATUS
approved
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