Search: a108524 -id:a108524
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A357585
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Triangle read by rows. Inverse of the convolution triangle of A108524, the number of ordered rooted trees with n generators.
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+20
2
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1, 0, 1, 0, 2, 1, 0, 7, 4, 1, 0, 32, 18, 6, 1, 0, 166, 92, 33, 8, 1, 0, 926, 509, 188, 52, 10, 1, 0, 5419, 2964, 1113, 328, 75, 12, 1, 0, 32816, 17890, 6792, 2078, 520, 102, 14, 1, 0, 203902, 110896, 42436, 13312, 3520, 772, 133, 16, 1
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OFFSET
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0,5
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COMMENTS
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Also the matrix inverse of the signed version of A105475 with 1, 0, 0, 0, ... as column 0.
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LINKS
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EXAMPLE
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Triangle T(n, k) starts:
[0] 1;
[1] 0, 1;
[2] 0, 2, 1;
[3] 0, 7, 4, 1;
[4] 0, 32, 18, 6, 1;
[5] 0, 166, 92, 33, 8, 1;
[6] 0, 926, 509, 188, 52, 10, 1;
[7] 0, 5419, 2964, 1113, 328, 75, 12, 1;
[8] 0, 32816, 17890, 6792, 2078, 520, 102, 14, 1;
[9] 0, 203902, 110896, 42436, 13312, 3520, 772, 133, 16, 1;
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MAPLE
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InvPMatrix := proc(dim, seqfun) local k, m, M, A;
if dim < 1 then return [] fi;
A := [seq(seqfun(i), i = 1..dim-1)];
M := Matrix(dim, shape=triangular[lower]); M[1, 1] := 1;
for m from 2 to dim do
M[m, m] := M[m - 1, m - 1] / A[1];
for k from m-1 by -1 to 2 do
M[m, k] := M[m - 1, k - 1] -
add(A[i+1] * M[m, k + i], i = 1..m-k) / A[1]
od od; M end:
InvPMatrix(10, n -> [1, -2][irem(n-1, 2) + 1]);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A108529
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Number of asymmetric mobiles (cycle rooted trees) with n generators.
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+10
10
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1, 1, 2, 5, 16, 51, 177, 621, 2246, 8245, 30783, 116257, 443945, 1710255, 6640939, 25961690, 102105115, 403701135, 1603721999, 6397931901, 25621989760, 102965680728, 415091909292, 1678226164646, 6803121058354, 27645628327636
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OFFSET
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1,3
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COMMENTS
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A generator is a leaf or a node with just one child.
Here CHK(A(x)) = 1 - Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)), i.e., the constant 1 is included in the definition of the CHK transform. For other sequences that involve the CHK transform, the 1 is sometimes dropped; e.g., see sequence A032171. We have CHK(A(x)) = x + x^2 + 3*x^3 + 8*x^4 + 27*x^5 + 86*x^6 + 303*x^7 + 1065*x^8 + 3871*x^9 + ... - Petros Hadjicostas, Dec 05 2017
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LINKS
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FORMULA
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G.f. satisfies: (2-x)*A(x) = x - 1 + CHK(A(x)).
a(n) = (1/2)*(a(n-1) + (1/n)*Sum_{d|n} mu(d)*c(n/d)) for n>=2, where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) and a(1) = c(1) = 1.
The g.f. satisfies (2-x)*A(x) = x - Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)). (This is just a rephrasing of C. Bower's equation above.)
The auxiliary sequence (c(n): n>=1} has g.f. C(x) = Sum_{n>=1} c(n)*x^n = x*(dA/dx)/(1-A(x)) = x + 3*x^2 + 10*x^3 + 35*x^4 + 136*x^5 + 528*x^6 + 2122*x^7 + ...
(End)
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PROG
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(PARI)
CHK(p, n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=x); for(n=2, n, p += x^n*polcoef(x*p + CHK(p, n), n)); Vecrev(p/x)} \\ Andrew Howroyd, Aug 31 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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STATUS
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approved
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A347205
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a(2n+1) = a(n) for n >= 0, a(2n) = a(n) + a(n - 2^A007814(n)) for n > 0 with a(0) = 1.
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+10
8
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1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 6, 3, 4, 1, 5, 4, 7, 3, 9, 5, 7, 2, 10, 6, 9, 3, 10, 4, 5, 1, 6, 5, 9, 4, 12, 7, 10, 3, 14, 9, 14, 5, 16, 7, 9, 2, 15, 10, 16, 6, 19, 9, 12, 3, 20, 10, 14, 4, 15, 5, 6, 1, 7, 6, 11, 5, 15, 9, 13, 4, 18, 12, 19, 7, 22, 10, 13
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OFFSET
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0,3
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COMMENTS
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Scatter plot might be called "Cypress forest on a windy day". - Antti Karttunen, Nov 30 2021
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LINKS
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FORMULA
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a(2n+1) = a(n) for n >= 0.
a(2n) = a(n) + a(n - 2^A007814(n)) = a(2*A059894(n)) for n > 0 with a(0) = 1.
Sum_{k=0..2^n - 1} a(k) = A000108(n+1) for n >= 0.
a((4^n - 1)/3) = A000108(n) for n >= 0.
a(2^m*(2^n - 1)) = binomial(n + m, n) for n >= 0, m >= 0.
Generalization:
b(2n+1, p, q) = b(n, p, q) for n >= 0.
b(2n, p, q) = p*b(n, p, q) + q*b(n - 2^A007814(n), p, q) = for n > 0 with b(0, p, q) = 1.
Sum_{k=0..2^n - 1} b(k, 2, 1) = A006318(n) for n >= 0.
Sum_{k=0..2^n - 1} b(k, 2, 2) = A115197(n) for n >= 0.
Sum_{k=0..2^n - 1} b(k, 3, 1) = A108524(n+1) for n >= 0.
Sum_{k=0..2^n - 1} b(k, 3, 3) = A116867(n) for n >= 0.
b((4^n - 1)/3, p, q) is generalized Catalan number C(p, q; n).
Conjecture: C(p, q; n) = Sum_{k=0..n-1} p^k*q^(n-k-1) Sum_{j=0..k} q^j*A009766(n-2, j) for n > 1 with C(p, q; 0) = C(p, q; 1) = 1.
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MATHEMATICA
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a[0] = 1; a[n_] := a[n] = If[OddQ[n], a[(n - 1)/2], a[n/2] + a[n/2 - 2^IntegerExponent[n/2, 2]]]; Array[a, 100, 0] (* Amiram Eldar, Sep 06 2021 *)
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PROG
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(PARI) a(n) = if (n==0, 1, if (n%2, a(n\2), a(n/2) + a(n/2 - 2^valuation(n/2, 2)))); \\ Michel Marcus, Sep 09 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A121575
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Riordan array (-sqrt(4*x^2+8*x+1)+2*x+2, (sqrt(4*x^2+8*x+1)-2*x-1)/2).
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+10
3
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1, -2, 1, 6, -5, 1, -24, 24, -8, 1, 114, -123, 51, -11, 1, -600, 672, -312, 87, -14, 1, 3372, -3858, 1914, -618, 132, -17, 1, -19824, 22992, -11904, 4218, -1068, 186, -20, 1, 120426, -140991, 75183, -28383, 8043, -1689, 249, -23, 1, -749976, 884112, -481704, 190347, -58398, 13929, -2508, 321, -26, 1
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OFFSET
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0,2
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COMMENTS
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First column is (-1)^n*A054872(n). Row sums are a signed version of A108524. Inverse of generalized Delannoy triangle A121574. Unsigned triangle is A121576.
Triangle T(n,k), 0 <= k <= n, read by rows, given by [ -2, -1, -3, -1, -3, -1, -3, -1, -3, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 09 2006
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LINKS
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FORMULA
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T(n,k) = (-1)^(n-k)*(1/2)*Sum_{i=0..n-k} binomial(n,i) * binomial(2*n-k-i,n)*(4 - 9*i + 3*i^2 - 6*(i-1)*n + 2*n^2)/((n-i+2)*(n-i+1))*2^i. - G. C. Greubel, Nov 02 2018
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EXAMPLE
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Triangle begins
1;
-2, 1;
6, -5, 1;
-24, 24, -8, 1;
114, -123, 51, -11, 1;
-600, 672, -312, 87, -14, 1;
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MATHEMATICA
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Flatten[Table[(-1)^(n-k)*Sum[Binomial[n, i] Binomial[2*n-k-i, n]*(4-9*i + 3*i^2 -6*(i-1)*n + 2*n^2)/((n-i+2)*(n-i+1))*2^i, {i, 0, n-k}]/2, {n, 0, 10}, {k, 0, n}]] (* G. C. Greubel, Nov 02 2018 *)
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PROG
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(PARI) for(n=0, 10, for(k=0, n, print1((-1)^(n-k)*sum(j=0, n-k, 2^j*binomial(n, j) *binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1)))/2, ", "))) \\ G. C. Greubel, Nov 02 2018
(Magma) [[(-1)^(n-k)*(&+[ 2^j*Binomial(n, j)*Binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1))/2: j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 02 2018
(GAP) T:=Flat(List([0..9], n->List([0..n], k->(-1)^(n-k)*Sum([0..n-k], i->Binomial(n, i)*Binomial(2*n-k-i, n)*(4-9*i+3*i^2-6*(i-1)*n+2*n^2)/((n-i+2)*(n-i+1))*2^i)/2))); # Muniru A Asiru, Nov 02 2018
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KEYWORD
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AUTHOR
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STATUS
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approved
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A108528
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Number of increasing mobiles (cycle rooted trees) with n generators.
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+10
2
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1, 2, 10, 92, 1216, 20792, 435520, 10793792, 308874016, 10021509632, 363509706880, 14576530558592, 640275236943616, 30573223563625472, 1576805482203235840, 87353392124392020992, 5173324070004374358016, 326160898887563325581312, 21810458629345555407462400
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OFFSET
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1,2
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COMMENTS
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In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.
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LINKS
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FORMULA
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E.g.f. satisfies 2*A(x) = x - 1 + A'(x) - log(1-A(x)).
E.g.f. satisfies: (1+A(x))*sqrt(1-A(x)^2) = exp(x).
E.g.f.: A(x) = Series_Reversion[ log((1+x)*sqrt(1-x^2)) ]. (End)
a(n) ~ 2^(n-2) * sqrt(3) * n^(n-1) / (exp(n) * (log(27/16))^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014
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MATHEMATICA
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Rest[CoefficientList[InverseSeries[Series[Log[(1+x)*Sqrt[1-x^2]], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)
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PROG
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(PARI) {a(n)=n!*polcoeff(serreverse(log((1+x)*sqrt(1-x^2+O(x^(n+2))))), n)} \\ Paul D. Hanna, Sep 11 2010
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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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A265435
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Riordan array (1, x*f(x)) where f(x) is the g.f. of A007564.
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+10
1
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1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 19, 9, 3, 1, 0, 100, 46, 15, 4, 1, 0, 562, 254, 82, 22, 5, 1, 0, 3304, 1476, 474, 128, 30, 6, 1, 0, 20071, 8893, 2847, 773, 185, 39, 7, 1, 0, 124996, 55046, 17587, 4796, 1165, 254, 49, 8, 1, 0, 793774, 347922, 111006, 30378, 7461, 1665, 336, 60, 9, 1
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OFFSET
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0,8
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COMMENTS
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Triangle T(n,k), 0<=k<=n, read by rows, given by (0, 1, 3, 1, 3, 1, 3, 1, 3, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
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LINKS
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EXAMPLE
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Triangle begins:
1
0, 1
0, 1, 1
0, 4, 2, 1
0, 19, 9, 3, 1
0, 100, 46, 15, 4, 1
Production matrix begins:
0, 1
0, 1, 1
0, 3, 1, 1
0, 9, 3, 1, 1
0, 27, 9, 3, 1, 1
0, 81, 27, 9, 3, 1, 1
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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