A090452 -id:A090452 - cp's OEIS frontend

A090442 Row sums of array A090452 (s2_{3,2}, scaled (3,2)-Stirling2).

1, 6, 44, 360, 3152, 28896, 273856, 2661504, 26380544, 265655808, 2710244352, 27952883712, 290977271808, 3053105307648, 32256844087296, 342870535471104, 3664053076557824, 39342496410894336, 424243929700630528, 4592400943255388160, 49885822426526253056

Offset: 1

Wolfdieter Lang, Dec 23 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.

Cf. A001003, A006318, A090452.

RecurrenceTable[{(n+1)*a[n] == 6*(2*n-1)*a[n-1] - 4*(n-2)*a[n-2],a[1]==1,a[2]==6},a,{n,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

a(n) = Sum_{m=2..2*n} A090452(n, m).
Recurrence: (n+1)*a(n) = 6*(2*n-1)*a(n-1) - 4*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ sqrt(4+3*sqrt(2))*(6+4*sqrt(2))^n/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
a(n) = 2^(n-1) * A001003(n) = 2^(n-2) * A006318(n). - Jacob Post, Jun 19 2018

A090453 Third column (m=4) of array A090452.

Original entry on oeis.org

2, 16, 51, 118, 230, 402, 651, 996, 1458, 2060, 2827, 3786, 4966, 6398, 8115, 10152, 12546, 15336, 18563, 22270, 26502, 31306, 36731, 42828, 49650, 57252, 65691, 75026, 85318, 96630, 109027, 122576, 137346, 153408, 170835, 189702, 210086

Offset: 0

Wolfdieter Lang, Dec 23 2003

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A055998, A090454.

Table[(n+1)(n^3+15n^2+56n+24)/12,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{2,16,51,118,230},40] (* Harvey P. Dale, Dec 07 2014 *)

a(n)= (n+1)*(n^3+15*n^2+56*n+24)/12.

G.f.: (2+6*x-9*x^2+3*x^3)/(1-x)^5.

A091026 Fifth column (m=6) of array A090452.

Original entry on oeis.org

5, 114, 771, 3235, 10365, 27825, 65828, 141552, 282375, 530090, 946275, 1619007, 2671123, 4270245, 6640800, 10078280, 14966001, 21794634, 31184795, 43912995, 60941265, 83450785, 112879860, 150966600, 199796675, 261856530, 340092459

Offset: 0

Wolfdieter Lang, Dec 23 2003

Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A090454, A091027.

a(n)= (n+1)*(n+2)*(n^6+45*n^5+745*n^4+5595*n^3+18694*n^2+22440*n+7200)/2880. 2880=8!/(2*7).

G.f.: (5+69*x-75*x^2-20*x^3+60*x^4-30*x^5+5*x^6)/(1-x)^9.

A090454 Fourth column (m=5) of array A090452.

Original entry on oeis.org

15, 105, 396, 1110, 2600, 5390, 10220, 18096, 30345, 48675, 75240, 112710, 164346, 234080, 326600, 447440, 603075, 801021, 1049940, 1359750, 1741740, 2208690, 2774996, 3456800, 4272125, 5241015, 6385680, 7730646, 9302910, 11132100

Offset: 0

Wolfdieter Lang, Dec 23 2003

Cf. A090453, A091026.

a(n)= (n+9)*(n+6)*(n+2)*(n+1)*(n^2+15*n+20)/144,

G.f.: (15-24*x^2+18*x^3-4*x^4)/(1-x)^7.

A091027 Sixth column (m=7) of array A090452.

Original entry on oeis.org

63, 910, 6083, 27483, 97188, 289884, 762048, 1816248, 4001613, 8264718, 16168867, 30204083, 54215070, 93980040, 157979640, 258402312, 412440315, 643939374, 985474539, 1480935379, 2188715144, 3185611044, 4571556360, 6475319760, 9061322985, 12537745038

Offset: 0

Wolfdieter Lang, Dec 23 2003

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A091026.

LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{63,910,6083,27483,97188,289884,762048,1816248,4001613,8264718,16168867},30] (* Harvey P. Dale, Dec 19 2021 *)

a(n)= (n+12)*(n+9)*(n+3)*(n+2)*(n+1)*(n^5+48*n^4+769*n^3+4782*n^2+11200*n+8400)/86400. 86400=10!/(2*3*7).

G.f.: (63+217*x-462*x^2+225*x^3+80*x^4-120*x^5+45*x^6-6*x^7)/(1-x)^11.

A091029 Signed array used for numerators of generating functions of the column sequences of array A090452.

Original entry on oeis.org

1, 3, -2, 2, 6, -9, 3, 15, 0, -24, 18, -4, 5, 69, -75, -20, 60, -30, 5, 63, 217, -462, 225, 80, -120, 45, -6, 14, 462, 300, -1848, 1785, -525, -210, 210, -63, 7, 252, 2460, -1809, -4932, 8428, -5208, 1050, 448, -336, 84, -8, 42, 2556, 9747, -18775, -2655, 28296, -28182, 12726, -1890, -840, 504, -108, 9

Offset: 2

Wolfdieter Lang, Dec 23 2003

The row polynomials P(m,x) := sum(a(m,k)*x^k,k=0..kmax(m)),m>=2, where kmax(m) := floor(3*m/2)-3=A032766(m-2)=[0,1,3,4,6,7,9,10,...], appear in the numerator of the g.f.s of the columns of A090452.

The sequence of the lengths of the rows is [1,2,4,5,7,8,10,11,13,14,...]=A001651(m-2)= floor((3*m-4)/2).

			[1]; [3,-2]; [2,6,-9,3]; [15,0,-24,18,-4]; ...
P(3,x)=3-2*x; P(5,x)=15-24*x^2+18*x^3-4*x^4.

W. Lang, First 9 rows.

a(m, k)=[x^k]P(m, x), with P(m, x) := ((1-x)^(2*m-3))*G(m, x)/x^ceiling(m/2) and the G(m, x) satisfy the hypergeometric differential difference eq. given in A090452.

A091031 Third to last entries in rows of array A090452 (scaled (3,2)-Stirling2).

Original entry on oeis.org

1, 16, 114, 644, 3270, 15642, 72072, 323752, 1428102, 6214520, 26761196, 114287736, 484816540, 2045277990, 8588492100, 35923392720, 149753379270, 622458753840, 2580709189740, 10675646755800, 44074384410420, 181638630107220, 747375951913344, 3070765657798704

Offset: 2

Wolfdieter Lang, Jan 09 2004

Cf. A000108 (last row entries), 3*A002054(n-1) (second to last row entries), A000108, A090452.

a[n_] := (9*n^2 - 19*n + 8)*(n-1) * CatalanNumber[n]/(4*(2*n-1)); Array[a, 25, 2] (* Amiram Eldar, Aug 30 2025 *)

a(n) = A090452(n, 2*(n-1)), n>=2.
a(n) = (9*n^2-19*n+8)*(n-1)*C(n)/(4*(2*n-1)), n>=2; with the Catalan numbers C(n) = A000108(n).
a(n) ~ 9 * 2^(2*n-3) * sqrt(n/Pi). - Amiram Eldar, Aug 30 2025

A078740 Triangle of generalized Stirling numbers S_{3,2}(n,k) read by rows (n>=1, 2<=k<=2n).

Original entry on oeis.org

1, 6, 6, 1, 72, 168, 96, 18, 1, 1440, 5760, 6120, 2520, 456, 36, 1, 43200, 259200, 424800, 285120, 92520, 15600, 1380, 60, 1, 1814400, 15120000, 34776000, 33566400, 16304400, 4379760, 682200, 62400, 3270, 90, 1, 101606400, 1117670400

Offset: 1

N. J. A. Sloane, Dec 21 2002

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

For the scaled array s2_{3,2}(n,k) := a(n,k)*k!/((n+1)!*n!) see A090452.

			1;
6, 6, 1;
72, 168, 96, 18, 1;
...

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (rows 1 <= n <= 100, flattened).
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
W. Lang, First 6 rows.
Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Row sums give A078738. Cf. A078739.

Cf. A005408, A090452.

a[n_, k_] := (-1)^k*n!*(n+1)!*HypergeometricPFQ[{2-k, n+1, n+2}, {2, 3}, 1]/(2*(k-2)!); Table[a[n, k], {n, 1, 7}, {k, 2, 2*n}] // Flatten (* Jean-François Alcover, Dec 04 2013 *)

Recursion: a(n, k) = Sum(binomial(2, p)*fallfac(n-1-p+k, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 1)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=3, s=2. 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)*(3-2), 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=3, s=2.
a(n, k) = (-1)^k n! (n+1)! 3F2(2-k, n+1, n+2; 2, 3; 1) / (2(k-2)!). - Jean-François Alcover, Dec 04 2013

Edited by Wolfdieter Lang, Dec 23 2003

cp's OEIS Frontend

A090442 Row sums of array A090452 (s2_{3,2}, scaled (3,2)-Stirling2).

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A090453 Third column (m=4) of array A090452.

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A091026 Fifth column (m=6) of array A090452.

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A090454 Fourth column (m=5) of array A090452.

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A091027 Sixth column (m=7) of array A090452.

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A091029 Signed array used for numerators of generating functions of the column sequences of array A090452.

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A091031 Third to last entries in rows of array A090452 (scaled (3,2)-Stirling2).

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A078740 Triangle of generalized Stirling numbers S_{3,2}(n,k) read by rows (n>=1, 2<=k<=2n).

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