A049424 -id:A049424 - cp's OEIS frontend

A049427 Row sums of triangle A049424.

1, 1, 5, 25, 145, 1025, 8245, 72745, 704705, 7424065, 83940805, 1012504505, 12972555025, 175624847425, 2501468566325, 37364323364425, 583569693556225, 9504040277271425, 161021013457176325, 2832196631069755225, 51619359912771959825

Offset: 0

Wolfdieter Lang

Seiichi Manyama, Table of n, a(n) for n = 0..510
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Column of A293991.
Row sums of A157394.
Cf. A005011.

E.g.f.: exp((-1+(1+x)^5)/5).
a(n) = n! * sum(k=0..n, sum(j=0..k, binomial(5*j,n) * (-1)^(k-j)/(5^k * (k-j)!*j!))). - Vladimir Kruchinin, Feb 07 2011
D-finite with recurrence a(n) -a(n-1) +4*(-n+1)*a(n-2) -6*(n-1)*(n-2)*a(n-3) -4*(n-1)*(n-2)*(n-3)*a(n-4) -(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jun 23 2023
a(n) = Sum_{k=0..n} Stirling1(n,k) * A005011(k). - Seiichi Manyama, Jan 31 2024
a(n) = (1/exp(1/5)) * n! * Sum_{k>=0} binomial(5*k,n)/(5^k * k!). - Seiichi Manyama, Jan 18 2025

A144878 Partition number array, called M31(-4), related to A049424(n,m) = S1(-4;n,m) (generalized Stirling triangle).

Original entry on oeis.org

1, 4, 1, 12, 12, 1, 24, 48, 48, 24, 1, 24, 120, 480, 120, 240, 40, 1, 0, 144, 1440, 1440, 360, 2880, 960, 240, 720, 60, 1, 0, 0, 2016, 10080, 504, 10080, 10080, 20160, 840, 10080, 6720, 420, 1680, 84, 1, 0, 0, 0, 16128, 20160, 0, 16128, 80640, 80640, 161280, 1344, 40320

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(-4;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=4) in the family M31(-K) of partition number arrays.
If M31(-4;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-4) := A049424.

			[1]; [4,1]; [12,12,1]; [24,48,48,24,1]; [24,120,480,120,240,40,1]; ...
a(4,3) = 48 = 3*S1(-4;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. A049427 (row sums).

Cf. A144877 (M31(-3) array), A144879 (M31(-5) array).

a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(S1(-4;j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S1(-4;j,1)^e(n,k,j),j=1..n) with S1(-4;n,1)= A008279(4,n-1)= [1,4,12,12,24,24,0,...], 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.

A157394 A partition product of Stirling_1 type [parameter k = 4] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 4, 1, 12, 12, 1, 72, 48, 24, 1, 280, 600, 120, 24, 1, 1740, 4560, 1800, 144, 0, 1, 8484, 40740, 21000, 2520, 0, 0, 1, 57232, 390432, 223440, 33600, 0, 0, 0, 1, 328752, 3811248, 2845584, 438480, 0, 0, 0, 0, 1, 2389140

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 = 4,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144878.
Same partition product with length statistic is A049424.
Diagonal a(A000217(n)) = falling_factorial(4,n-1), row in A008279
Row sum is A049427.

			1
1       4
1      12       12
1      72       48       24
1     280      600      120      24
1    1740     4560     1800     144  0
1    8484    40740    21000    2520  0  0
1   57232   390432   223440   33600  0  0  0
1  328752  3811248  2845584  438480  0  0  0  0
1  2389140

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+6).

cp's OEIS Frontend

A049427 Row sums of triangle A049424.

Views

Author

Keywords

Links

Crossrefs

Formula

A144878 Partition number array, called M31(-4), related to A049424(n,m) = S1(-4;n,m) (generalized Stirling triangle).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A157394 A partition product of Stirling_1 type [parameter k = 4] with biggest-part statistic (triangle read by rows).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula