A134144 -id:A134144 - cp's OEIS frontend

A049118 Row sums of triangle A035342 and array A134144.

1, 4, 25, 211, 2236, 28471, 422899, 7173580, 136750051, 2893057381, 67241818876, 1702829138209, 46659181547785, 1375237342827076, 43380198327693361, 1458027134026128691, 52014149849253158284, 1962794208713975883415

Offset: 1

Wolfdieter Lang

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2002.
P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, Boson normal ordering via substitutions and Sheffer-type polynomials, arXiv:quant-ph/0501155, 2005.

Cf. A000262, A049119, A049120.

a[n_, k_] := 2^(n+k)*n!/(4^n*n*k!)*Sum[(j+k)*2^(j)*Binomial[j+k-1, k-1]*Binomial[2*n-j-k-1, n-1], {j, 0, n-k}]; a[n_] := Sum[a[n, k], {k, 1, n}]; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Jul 05 2013, after Emanuele Munarini *)
Table[Sum[BellY[n, k, (2 Range[n] - 1)!!], {k, n}], {n, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

a(n,k):=2^(n+k)*n!/(4^n*n*k!)*sum((j+k)*2^(j)*binomial(j+k-1, k-1)*binomial(2*n-j-k-1, n-1), j, 0, n-k); makelist(sum(a(n,k),k,0,n),n,1,12); /* Emanuele Munarini, Jun 01 2012 */

E.g.f. exp(-1+1/sqrt(1-2*x))-1.
Representation of a(n) as n-th moment of a positive function on (0, infinity): a(n)=int(x^n* (x/2)^(-1/2)*exp(-x/2)*(2*hypergeom([], [3/2, 1/2], 1/8*x)/Pi^(1/2)+1/2*sqrt(2)*sqrt(x)*hypergeom([], [2, 3/2], 1/8*x))/(4*exp(1)), x=0..infinity), n=1, 2, ... - Karol A. Penson, Jun 27 2002
Asymptotic expansion for large n: a(n) -> 2^(1/6)*(n^(-1/3) + 2^(-7/3)*n^(-2/3) + O(1/n))*(2*n)^n*exp(-n+(3/2)*(2*n)^(1/3))/(sqrt(3)*exp(1)); (the nature of this approximation of a(n) is the same as that of Stirling approximation of n!). - Karol A. Penson, Sep 02 2002
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+x)^3*d/dx. Cf. A000110, A000262, A049119 and A049120. - Peter Bala, Nov 25 2011

A157399 A partition product of Stirling_2 type [parameter k = -3] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 3, 1, 9, 15, 1, 45, 60, 105, 1, 165, 600, 525, 945, 1, 855, 5250, 6300, 5670, 10395, 1, 3843, 39900, 91875, 79380, 72765, 135135, 1, 21819, 391440, 1164975, 1323000, 1164240, 1081080, 2027025, 1

Offset: 1

Peter Luschny, Mar 09 2009, Mar 14 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -3,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A134144.
Same partition product with length statistic is A035342.
Diagonal a(A000217) = A001147.
Row sum is A049118.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157396, A157397, A157398, A157400, A080510, A157401, A157402, A157403, A157404, A157405

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(-2*j - 1).

A134145 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(3)/M_3.

Original entry on oeis.org

1, 3, 1, 15, 3, 1, 105, 15, 9, 3, 1, 945, 105, 45, 15, 9, 3, 1, 10395, 945, 315, 225, 105, 45, 27, 15, 9, 3, 1, 135135, 10395, 2835, 1575, 945, 315, 225, 135, 105, 45, 27, 15, 9, 3, 1, 2027025, 135135, 31185, 14175, 11025, 10395, 2835, 1575, 945, 675, 945, 315, 225

Offset: 1

Wolfdieter Lang, Nov 13 2007

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_3(3) = A134144 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short M_3(3)/M_3.

			[1]; [3,1]; [15,3,1]; [105,15,9,3,1]; [945,105,45,15,9,3,1]; ...
a(4,3)=9 from the third (k=3) partition (2^2) of 4: (3)^2 = 9, because S2(3,2,1) = 3!! = 1*3 = 3.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Cf. A134147 (row sums, also of triangle A134146).

a(n,k) = Product_{j=1..n} S2(3,j,1)^e(n,k,j) with S2(3,n,1) = A035342(n,1) = A001147(n) = (2*n-1)!! and with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

a(n,k) = A134144(n,k)/A036040(n,k) (division of partition arrays M_3(3) by M_3).

A134149 A certain partition array in Abramowitz-Stegun (A-St) order.

Original entry on oeis.org

1, 4, 1, 28, 12, 1, 280, 112, 48, 24, 1, 3640, 1400, 1120, 280, 240, 40, 1, 58240, 21840, 16800, 7840, 4200, 6720, 960, 560, 720, 60, 1, 1106560, 407680, 305760, 274400, 76440, 117600, 54880, 47040, 9800, 23520, 6720, 980, 1680, 84, 1, 24344320

Offset: 1

Wolfdieter Lang, Nov 13 2007

For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_3(4), the k=4 member of a family of generalizations of the multinomial number array M_3 = M_3(1) = A036040.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
The S2(4,n,m) numbers (generalized Stirling2 numbers) are obtained by summing in row n all numbers with the same part number m. In the same manner the S2(n,m) (Stirling2) numbers A008277 are obtained from the partition array M_3= A036040.
a(n,k) enumerates unordered forests of increasing quaternary trees related to the k-th partition of n in the A-St order. The m-forest is composed of m such trees, with m the number of parts of the partition.

			[1]; [4,1]; [28,12,1]; [280,112,48,24,1]; [3640,1400,1120,280,240,40,1]; ...
a(4,3)=48 from the third (k=3) partition (2^2) of 4: 4!*((4/2!)^2)/2 = 48, because S2(4,2,1) = 4!!! = 4*1 = 4.
There are a(4,3) = 48 = 3*4^2 unordered 2-forests with 4 vertices, composed of two increasing quaternary (4-ary) trees, each with two vertices: there are 3 increasing labelings (1,2)(3,4); (1,3)(2,4); (1,4)(2,3) and each tree comes in four versions from the quaternary structure.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Cf. A134144 (M_3(3) array).

a(n,k) = n!*Product_{j=1..n} (S2(4,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(4,n,1) = A035469(n,1) = A007559(n) = (3*n-2)!!! (triple- or 3-factorials) and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1.

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A049118 Row sums of triangle A035342 and array A134144.

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A157399 A partition product of Stirling_2 type [parameter k = -3] with biggest-part statistic (triangle read by rows).

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A134145 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(3)/M_3.

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A134149 A certain partition array in Abramowitz-Stegun (A-St) order.

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