A124373 -id:A124373 - cp's OEIS frontend

A135920 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k^2*x).

1, 1, 2, 7, 37, 264, 2433, 27913, 386906, 6346119, 121159373, 2655174768, 66028903633, 1845579100993, 57506847262162, 1983312152411351, 75238783332550789, 3122408658986242072, 141063757638078429489

Offset: 0

Paul D. Hanna, Dec 06 2007

nonn

From Peter Bala, Sep 27 2012: (Start)
Generalized Bell numbers; row sums of A036969.
a(n) is equal to the number of partitions of the set {1,1',2,2',...,n,n'} into disjoint nonempty subsets V1,...,Vk (1 <= k <= n) such that, for each 1 <= j <= k, if i is the least integer such that either i or i' belongs to Vj then {i,i'} is a subset of Vj.
Example: a(3) = 7: There is a single partition into one set {1,1',2,2',3,3'}; five partitions into two sets, namely, {1,1',2,2'}{3,3'}, {1,1',3,3'}{2,2'}, {1,1'}{2,2',3,3'}, {1,1',3}{2,2',3'} and {1,1',3'}{2,2',3}; and finally a single partition into three sets {1,1'}{2,2'}{3,3'}. (End)

			O.g.f.: A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-4x)) + x^3/((1-x)*(1-4x)*(1-9x)) + x^4/((1-x)*(1-4x)*(1-9x)*(1-16x)) + ...
Also generated by iterated binomial transforms in the following way:
[1,2,7,37,264,2433,27913,...] = BINOMIAL([1,1,4,21,151,1422,16629,..]);
[1,4,21,151,1422,16629,234529,...] = BINOMIAL^3([1,1,6,43,393,4596,...]);
[1,6,43,393,4596,66049,1125905,...] = BINOMIAL^5([1,1,8,73,811,11274,...]);
[1,8,73,811,11274,191685,...] = BINOMIAL^7([1,1,10,111,1453,23328,...]);
[1,10,111,1453,23328,456033,...] = BINOMIAL^9([1,1,12,157,2367,43014,...]);
etc.

S. Matsumoto and J. Novak, Jucys-Murphy Elements and Unitary Matrix Integrals arXiv.0905.1992 [math.CO], 2009-2012.

Cf. A135921, A124373, A000110, A036969.

nmax = 20;
A[x_] = Sum[x^n/Product[1 - k^2 x, {k, 0, n}], {n, 0, nmax}];
CoefficientList[A[x] + O[x]^nmax, x] (* Jean-François Alcover, Jul 27 2018 *)

a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-j^2*x+x*O(x^n))), n)

From Peter Bala, Sep 27 2012: (Start)
Let E(x) = cosh(sqrt(2*x)) = Sum_{n >= 0} x^n/((2*n)!/2^n). A generating function is E((E(x) - 1)) = 1 + x + 2*x^2/6 + 7*x^3/90 + ..., where the sequence of denominators [1, 1, 6, 90, ...] is given by (2*n)!/2^n. Cf. A000110 which has generating function exp((exp(x) - 1)).
An e.g.f. is E((E(x^2/2) - 1)) = 1 + x^2/2! + 2*x^4/4! + 7*x^6/6! + .... (End)
G.f.: 1 + x/(U(0)-x) where U(k) = 1 - 2*x*k - x*k^2 + x*(x*(k+1)^2 - 1)/U(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Oct 11 2012
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-k^2*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
Conjecture: a(n) = A369527(n-1, 0) = A369595(n-1, 0) for n > 0 with a(0) = 1. - Mikhail Kurkov, Apr 25 2024

A080248 Stirling-like number triangle defined by sequence A000217.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 13, 10, 1, 1, 40, 73, 20, 1, 1, 121, 478, 273, 35, 1, 1, 364, 2989, 3208, 798, 56, 1, 1, 1093, 18298, 35069, 15178, 1974, 84, 1, 1, 3280, 110881, 368988, 262739, 56632, 4326, 120, 1, 1, 9841, 668566, 3800761, 4310073, 1452011, 177760

Offset: 0

Paul Barry, Feb 17 2003

Columns include A003462, A016211, A021514. The defining sequence A000217(n) = C(n+1,2) is the sequence of partial sums of the sequence (0,1,2,3,4,...) which defines the Stirling numbers of the second kind A008277.
n-th row = M^n * [1,0,0,0,...], where M = an infinite lower triangular matrix with (1, 3, 6, ...) in the main diagonal and (1, 1, 1, ...) in the subdiagonal. - Gary W. Adamson, Apr 13 2009
Row sums = A124373 starting (1, 2, 6, 25, 135, ...). - Gary W. Adamson, Jul 11 2011

			Rows are
  {1},
  {1,  1},
  {1,  4,  1},
  {1, 13, 10,  1},
  {1, 40, 73, 20, 1},
  ...
For example, 73 = 13 + 6*10, 20 = 10 + 10*1.

Vincenzo Librandi, Rows n = 0..50, flattened

Cf. A000217, A008277, A124373.

gf  := k -> 1/mul(1 - x*j*(j-1)/2, j=0..k+2):
ser := k -> series(gf(k), x, 16):
T := (n, k) -> coeff(ser(k), x, n-k):
seq(print(seq(T(n, k), k=0..n)), n=0..8); # Peter Luschny, Aug 29 2020

max = 10; t[n_, n_] = n*(n+1)/2; t[n_, k_] /; k == n-1 = 1; t[, ] = 0; m = Table[t[n, k], {n, 1, max}, {k, 1, max}]; row[n_] := MatrixPower[m, n][[All, 1]]; Table[Take[row[n], n+1], {n, 0, max-1}] // Flatten (* Jean-François Alcover, Jun 25 2013, after Gary W. Adamson *)

{T(n, k) = local(s); if( k<0 || k>n, 0, forvec(v = vector(n-k, i, [0, k]), s += prod(i=1, n-k, v[i] * (v[i] + 1) / 2), 1)); s}; /* Michael Somos, Feb 06 2004 */

Columns are generated by 1/Product_{k=1..n+1} (1 - C(k + 1, 2)*x). [In other words:
T(n, k) = [x^(n-k)] 1/Product_{j=0..k+2}(1 - x*binomial(j, 2)).]
T(n, k) = (k*(k+1)/2) * T(n-1,k) + T(n-1,k-1), T(n,n)=1. - Vladimir Kruchinin, Aug 25 2020
T(n,k) = (Sum_{i=0..k} (-1)^(k-i) * (2*i + 3) * binomial(2*k + 3,k-i) * ((i+1) * (i+2) / 2)^(n+1)) * 2^(k+1) / (2*k + 3)! for 0 <= k <= n. - Werner Schulte, Oct 29 2020
The polynomials p(n,x) = Sum_{k=0..n} T(n,k) * (k!*(k+1)!/2^k) * x^(k+2) satisfy for n >= 0 the equations p(n+1,x) = p(1,x) * p''(n,x) / 2 and p(n,-1) = 0^n when p'' is the second derivative of p. - Werner Schulte, Dec 15 2020

A135921 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k(k+1)x).

Original entry on oeis.org

1, 1, 3, 13, 81, 669, 6955, 88505, 1346209, 23998521, 493956467, 11596542533, 307301505073, 9110471500693, 299893197116059, 10888674034993905, 433549376981078593, 18833037527449398129, 888439543634687700579

Offset: 0

Paul D. Hanna, Dec 06 2007

nonn

			O.g.f.: A(x) = 1 + x/(1-2x) + x^2/((1-2x)*(1-6x)) + x^3/((1-2x)*(1-6x)*(1-12x)) + x^4/((1-2x)*(1-6x)*(1-12x)*(1-20x)) + ...
Also generated by iterated binomial transforms in the following way:
[1,3,13,81,669,6955,88505,...] = BINOMIAL^2([1,1,5,31,253,2673,34833,..]);
[1,5,31,253,2673,34833,541879,...] = BINOMIAL^4([1,1,7,57,577,7389,...]);
[1,7,57,577,7389,115983,2151493,...] = BINOMIAL^6([1,1,9,91,1101,16497,...]);
[1,9,91,1101,16497,301669,..] = BINOMIAL^8([1,1,11,133,1873,32061,..]);
[1,11,133,1873,32061,666579,...] = BINOMIAL^10([1,1,13,183,2941,56529,...]);
etc.

Cf. A071951, A135920, A124373.

a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-j*(j+1)*x+x*O(x^n))), n)

{a(n) = sum( k=0, n, sum( i=0, k, (-1)^(i+k) * (2*i + 1) * (i*i + i)^n / (k-i)! / (k+i+1)! ))} /* Michael Somos, Feb 25 2012 */

a(n+1) = row sums of A071951. - Michael Somos, Feb 25 2012

G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-(k+1)*(k+2)*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013

cp's OEIS Frontend

A135920 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k^2*x).

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A080248 Stirling-like number triangle defined by sequence A000217.

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A135921 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k(k+1)x).

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cp's OEIS Frontend

A135920 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k^2*x).

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A080248 Stirling-like number triangle defined by sequence A000217.

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Maple

Mathematica

PARI

Formula

A135921 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k*(k+1)*x).

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A135921 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k(k+1)x).