A049411 -id:A049411 - cp's OEIS frontend

A049428 Row sums of triangle A049411.

1, 1, 6, 36, 246, 2046, 19716, 209616, 2441916, 31050396, 425883816, 6244077456, 97391939976, 1609040166696, 28029696862896, 512903202039936, 9829166157390096, 196739739722616336, 4102788435212513376, 88945209649582514496, 2000700796384204930656

Offset: 0

Wolfdieter Lang

Seiichi Manyama, Table of n, a(n) for n = 0..485
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Column k=5 of A293991.

Cf. A005012.

nmax = 20;
a[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, nmax}]];
a[0] = 1; a[n_] := Sum[a[n, m], {m, 1, n}];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jul 27 2018 *)

E.g.f.: exp((-1+(1+x)^6)/6).
a(n) = n! * Sum_{k=1..n} Sum_{j=0..k} binomial(6*j,n) *(-1)^(k-j)/ (6^k*(k-j)!*j!). - Vladimir Kruchinin, Feb 07 2011
D-finite with recurrence a(n) -a(n-1) +5*(-n+1)*a(n-2) -10*(n-1)*(n-2)*a(n-3) -10*(n-1)*(n-2)*(n-3)*a(n-4) -5*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-6)=0. - R. J. Mathar, Jun 23 2023
a(n) = Sum_{k=0..n} Stirling1(n,k) * A005012(k). - Seiichi Manyama, Jan 31 2024
a(n) = (1/exp(1/6)) * n! * Sum_{k>=0} binomial(6*k,n)/(6^k * k!). - Seiichi Manyama, Jan 18 2025

Offset adjusted by R. J. Mathar, Aug 29 2009

A144879 Partition number array, called M31(-5), related to A049411(n,m) = S1(-5;n,m) (generalized Stirling triangle).

Original entry on oeis.org

1, 5, 1, 20, 15, 1, 60, 80, 75, 30, 1, 120, 300, 1000, 200, 375, 50, 1, 120, 720, 4500, 4000, 900, 6000, 1875, 400, 1125, 75, 1, 0, 840, 12600, 42000, 2520, 31500, 28000, 52500, 2100, 21000, 13125, 700, 2625, 105, 1, 0, 0, 16800, 134400, 126000, 3360, 100800, 336000

Offset: 1

Wolfdieter Lang Oct 09 2008, Oct 28 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31(-5;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
First member (K=5) in the family M31(-K) of partition number arrays.
If M31(-5;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-5) := A049411.

			[1]; [5,1]; [20,15,1]; [60,80,75,30,1]; [120,300,1000,200,375,50,1]; ...
a(4,3) = 75 = 3*S1(-5;2,1)^2. The relevant partition of 4 is (2^2).

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Cf. A049428 (row sums).

Cf. A144878 (M31(-4) array).

a(n,k)=(n!/Product_{j=1..n} (e(n,k,j)!*j!^e(n,k,j))) * Product_{j=1..n} S1(-5;j,1)^e(n,k,j) = M3(n,k) * Product_{j=1..n} S1(-5;j,1)^e(n,k,j), with S1(-5;n,1) = A008279(5,n-1)= [1,5,20,60,120,120,0,...], n>=1 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. M3(n,k)=A036040.

A157395 A partition product of Stirling_1 type [parameter k = 5] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 5, 1, 15, 20, 1, 105, 80, 60, 1, 425, 1200, 300, 120, 1, 3075, 10400, 5400, 720, 120, 1, 15855, 102200, 75600, 15120, 840, 0, 1, 123515, 1149120, 907200, 241920, 20160, 0, 0, 1, 757755, 12783680, 13426560, 3719520, 362880

Offset: 1

Peter Luschny, Mar 07 2009, Mar 14 2009

Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = 5,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144879.
Same partition product with length statistic is A049411.
Diagonal a(A000217(n)) = falling_factorial(5,n-1), row in A008279
Row sum is A049428.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_1 Triangles.

Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395

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-2}(j-n+7).

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A049428 Row sums of triangle A049411.

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A144879 Partition number array, called M31(-5), related to A049411(n,m) = S1(-5;n,m) (generalized Stirling triangle).

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A157395 A partition product of Stirling_1 type [parameter k = 5] with biggest-part statistic (triangle read by rows).

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