A090438 -id:A090438 - cp's OEIS frontend

A092077 Generalized Stirling2 array (8,2).

1, 56, 16, 1, 10192, 4928, 776, 48, 1, 3872960, 2477440, 575680, 63360, 3536, 96, 1, 2517424000, 1940556800, 572868800, 86163840, 7326880, 364800, 10480, 160, 1, 2497284608000, 2210343116800, 773352966400, 143430604800, 15836206400, 1099612800, 49056960, 1398400, 24520, 240, 1

Offset: 1

Wolfdieter Lang, Feb 27 2004

The sequence of row lengths for this array is [1,3,5,7,9,11,...]=A005408(n-1), n>=1.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Wolfdieter Lang, First 6 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

The generalized (k, 2)-Stirling2 arrays are, for k=2, ..., 7: A078739, A078740, A090438, A091534, A091746 and A091747.

Cf. A091546, A091552 (first, resp. second column). A091757 (row sums). A091758 (alternating row sums).

a[n_, k_] := ((-1)^k/k!) Sum[(-1)^p Binomial[k, p] Product[FactorialPower[ p + 6(j-1), 2], {j, 1, n}], {p, 2, k}];
Table[a[n, k], {n, 1, 6}, {k, 2, 2n}] // Flatten (* Jean-François Alcover, Feb 28 2020 *)

a(n, k) = (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+6*(j-1), 2), j=1..n), p=2..k), n>=1, 2<=k<=2*n, else 0. From eq. (12) of the Blasiak et al. reference with r=8, s=2.

Recursion: a(n, k) = sum(binomial(2, p)*fallfac(6*(n-1)+k-p, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 2)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=8, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

A090440 Generalized Stirling2 array (4,3).

Original entry on oeis.org

1, 24, 36, 12, 1, 1440, 5760, 6120, 2520, 456, 36, 1, 172800, 1339200, 2808000, 2420640, 1025280, 232920, 29400, 2040, 72, 1, 36288000, 471744000, 1643846400, 2381702400, 1745755200, 721224000, 178941600, 27624960, 2689920, 163800, 6000

Offset: 1

Wolfdieter Lang, Dec 23 2003

The row lengths for this array are [1,4,7,10,13,16,...] = A016777(n-1), n>=1.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

W. Lang, First 6 rows.

Cf. A090438 (4, 2)-Stirling2.

Cf. A070531 (row sums), A091028 (alternating row sums).

ff[n_, k_] = Pochhammer[n - k + 1, k]; a[1, 3] = 1; a[n_, k_] := a[n, k] = Sum[Binomial[3, p]*ff[(n - 1 - p + k), 3 - p]*a[n - 1, k - p], {p, 0, 3} ]; a[n_ /; n < 2, ] = 0; Flatten[Table[a[n, k], {n, 1, 5}, {k, 3, 3 n}]] (* _Jean-François Alcover, Sep 01 2011, after given recursion *)

Recursion: a(n, k)=sum(binomial(3, p)*fallfac(n-1-p+k, 3-p)*a(n-1, k-p), p=0..3), n>=2, 3<=k<=3*n, a(1, 3)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=4, s=3. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+j-1, 3), j=1..n), p=3..k), n>=1, 3<=k<=3*n, else 0. From eq. (12) of the Blasiak et al. reference with r=4, s=3.

A072678 Generalized Bell numbers B_{4,2}.

Original entry on oeis.org

1, 21, 1045, 93289, 12975561, 2581284541, 693347907421, 241253367679185, 105394372192969489, 56410454014314490981, 36271084122927079387941, 27567930377271475039277881, 24435533594428382909107147225

Offset: 1

Karol A. Penson, Jul 01 2002

nonn

Robert Israel, Table of n, a(n) for n = 1..220
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem., arXiv:quant-phys/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A090439 (alternating row sums of A090438).

f:= n -> simplify((2*n)!*hypergeom([2*n+1], [3], 1)/(2*exp(1))):
map(f, [$1..30]); # Robert Israel, May 23 2016

a[n_] := n*(2n-1)!*Hypergeometric1F1[2-2n, 3, -1]; Array[a, 30] (* Jean-François Alcover, Sep 01 2016 *)

a(n) = (2*n)!*hypergeom([2*n+1], [3], 1)/(2*exp(1)), n=1, 2, ... Special values of the confluent hypergeometric function 1F1.
a(n) = sum(A090438(n, k), k=2..2*n)= sum((1/k!)*product(fallfac(k+(j-1)*(4-2), 2), j=1..n), k=1..infinity)/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=4, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.
8*n*(2*n-1)*(2*n+1)*(n+1)^2*(n+3)*(n+2)*a(n)+(2*(n+1))*(8*n^3+32*n^2+42*n+13)*a(n+1)*(n+3)*(n+2)-(8*n^2+38*n+51)*(n+3)*(n+2)*a(n+2)+(n+3)*(n+2)*a(n+3) = 0. - Robert Israel, May 23 2016
a(n) = A052852(2*n-1). - Mark van Hoeij, Sep 05 2022

Edited by Wolfdieter Lang, Dec 23 2003

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A092077 Generalized Stirling2 array (8,2).

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A090440 Generalized Stirling2 array (4,3).

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A072678 Generalized Bell numbers B_{4,2}.

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