A091746 -id:A091746 - cp's OEIS frontend

A091550 Second column (k=3) sequence of array A091746 ((6,2)-Stirling2) divided by 12.

1, 160, 39900, 15120000, 8202070800, 6058891238400, 5860547004312000, 7196668193594880000, 10944624305020966560000, 20199809308312018344960000, 44490168120726255724917120000, 115290834599202214240544256000000, 347284815748143369922163257920000000

Offset: 2

Wolfdieter Lang, Feb 13 2004

Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; arXiv preprint, arXiv:quant-ph/0402027, 2004.

Cf. A091539 (second column of (5, 2)-Stirling2 array), A091550 (second column of (7, 2)-Stirling2 array), A091746.

a[n_] := 2^(4*n) * Pochhammer[1/2, n] * (-3 * Pochhammer[1/4, n] + Pochhammer[3/4, n])/(3!*12); Array[a, 20, 2] (* Amiram Eldar, Aug 30 2025 *)

a(n) = (2^(4*n)) * risefac(1/2, n) * (-3*risefac(1/4, n) + risefac(3/4, n))/(3!*12), n>=2, with risefac(x, n) = Pochhammer(x, n).
E.g.f.: (hypergeom([1/2, 3/4], [], 16*x) - 3*hypergeom([1/4, 1/2], [], 16*x) + 2)/(3!*12).
a(n) = (2^n) * Product_{j=0..n-1} (2*j+1) * (-3*Product_{j=0..n-1} (4*j+1) + Product_{j=0..n-1} (4*j+3))/(3!*12), n>=2. From eq.12 of the Blasiak et al. reference with r=6, s=2, k=3.
a(n) ~ sqrt(Pi) * 2^(4*n-2) * n^(2*n+1/4) / (9 * Gamma(3/4) * exp(2*n)). - Amiram Eldar, Aug 30 2025

A091544 First column sequence of array A091746 ((6,2)-Stirling2).

Original entry on oeis.org

1, 30, 2700, 491400, 150368400, 69470200800, 45155630520000, 39285398552400000, 44078217175792800000, 61973973349164676800000, 106719182107261573449600000, 220908706962031457040672000000, 541226332056977069749646400000000, 1548989762347068373623487996800000000

Offset: 1

Wolfdieter Lang, Feb 13 2004

Also fifth column (m=4) sequence of triangle A091543.

Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; arXiv preprint, arXiv:quant-ph/0402027, 2004.

Cf. A091535 (third column of A091543, first column of array A091534), A000407, A007696, A091746.

a[n_] := 2^(4*n-1) * Pochhammer[1/4, n] * Pochhammer[1/2, n]; Array[a, 20] (* Amiram Eldar, Aug 30 2025 *)

a(n) = 2^(n-1)*Product_{j=0..n-1}((2*j+1)*(4*j+1)), n>=1. From eq.12 of the Blasiak et al. reference with r=6, s=2, k=1.
a(n) = (2^(4*n-1))*risefac(1/4, n)*risefac(1/2, n), n>=1, with risefac(x, n) = Pochhammer(x, n).
a(n) = fac4(4*n-3)*fac4(4*n-2)/2, n>=1, with fac4(4*n-3) = A007696(n) and fac4(4*n-2)/2 = A000407(n+1) (quartic- or 4-factorials).
E.g.f.: (hypergeom([1/4, 1/2], [], 16*x)-1)/2.
a(n) = A091746(n, 2), n>=1.
a(n) ~ sqrt(Pi) * 2^(4*n) * n^(2*n-1/4) / (Gamma(1/4) * exp(2*n)). - Amiram Eldar, Aug 30 2025

A091750 Alternating row sums of array A091746 (generalized Stirling2 array (6,2)).

Original entry on oeis.org

1, 19, 1171, 151681, 33743041, 11461272931, 5499378001699, 3524209981309921, 2887597622445729121, 2916075074262560119891, 3507033649107862877516371, 4836904652902464306386484769, 7251193844710136332044143616481

Offset: 1

Wolfdieter Lang, Feb 27 2004

The first negative term is the 41 digit number a(16)=-42686874251433290891296448926053802610879.

Cf. A091748 (row sums of A091746).

a(n)=sum(A091746(n, k)*(-1)^k, k=2..2*n), n>=1.

A091747 Generalized Stirling2 array (7,2).

Original entry on oeis.org

1, 42, 14, 1, 5544, 3192, 588, 42, 1, 1507968, 1165248, 321552, 41496, 2688, 84, 1, 696681216, 655966080, 232606080, 41497344, 4143552, 240240, 7980, 140, 1, 489070213632, 533531142144, 226306918656, 50249808000, 6575950080

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, 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. A078740 (3, 2)-Stirling2, A090438 (4, 2)-Stirling2, A091534 (5, 2)-Stirling2, A091746 (6, 2)-Stirling2.
Cf. A091545 (first column).
Cf. A091749 (row sums), A091751 (alternating row sums).

a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+5*(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=7, s=2.

Recursion: a(n, k)=sum(binomial(2, p)*fallfac(5*(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=7, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

A092077 Generalized Stirling2 array (8,2).

Original entry on oeis.org

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).

A091748 Generalized Bell numbers B_{6,2}.

Original entry on oeis.org

1, 43, 5083, 1160113, 432168721, 238012552651, 181520958432283, 182989529196234433, 235492729726705299073, 376560458072018837889931, 732162019709408940671604091, 1700645336651586566571229542193

Offset: 1

Wolfdieter Lang, Feb 27 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. A072019 ( B_{5, 2}).

a[n_] := Sum[Product[FactorialPower[k+4*(j-1), 2], {j, 1, n}]/k!, {k, 2, Infinity}]/E; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Sep 01 2016 *)

a(n)=sum(A091746(n, k), k=2..2*n)= sum((1/k!)*product(fallfac(k+4*(j-1), 2), j=1..n), k=2..infinity)/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=6, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.

cp's OEIS Frontend

A091550 Second column (k=3) sequence of array A091746 ((6,2)-Stirling2) divided by 12.

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A091544 First column sequence of array A091746 ((6,2)-Stirling2).

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A091750 Alternating row sums of array A091746 (generalized Stirling2 array (6,2)).

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A091747 Generalized Stirling2 array (7,2).

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A092077 Generalized Stirling2 array (8,2).

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A091748 Generalized Bell numbers B_{6,2}.

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