A011801 -id:A011801 - cp's OEIS frontend

A157404 A partition product of Stirling_2 type [parameter k = 4] with biggest-part statistic (triangle read by rows).

1, 1, 4, 1, 12, 36, 1, 72, 144, 504, 1, 280, 1800, 2520, 9576, 1, 1740, 22320, 37800, 57456, 229824, 1, 8484, 182700, 864360, 1005480, 1608768, 6664896, 1, 57232, 2380896, 16546320, 26276544, 32175360, 53319168, 226606464

Offset: 1

Peter Luschny, Mar 09 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 4,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144267.
Same partition product with length statistic is A011801.
Diagonal a(A000217) = A008546.
Row sum is A028575.

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

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, 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}(5*j - 1).

A144285 Lower triangular array called S2hat(-4) related to partition number array A144284.

Original entry on oeis.org

1, 4, 1, 36, 4, 1, 504, 52, 4, 1, 9576, 648, 52, 4, 1, 229824, 12888, 712, 52, 4, 1, 6664896, 286272, 13464, 712, 52, 4, 1, 226606464, 8182944, 299520, 13720, 712, 52, 4, 1, 8837652096, 266366016, 8455392, 301824, 13720, 712, 52, 4, 1, 388856692224, 10191545280, 273091392

Offset: 1

Wolfdieter Lang Oct 09 2008

If in the partition array M32khat(-4)= A144284 entries with the same parts number m are summed one obtains this triangle of numbers S2hat(-4). In the same way the Stirling2 triangle A008277 is obtained from the partition array M_3 = A036040.

The first three columns are A008546, A144339, A144340.

			[1];[4,1];[36,4,1];[504,52,4,1];[9576,648,52,4,1];...

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.

Row sums A144286.

A144280 (S2hat(-3)), A144342 (S2hat(-5)).

a(n,m)=sum(product(|S2(-4;j,1)|^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. |S2(-4,n,1)|= A011801(n,1) = A008546(n-1) = (5*n-6)(!^5) (5-factorials) for n>=2 and 1 if n=1.

A049424 Triangle read by rows, the Bell transform of n!*binomial(4,n) (without column 0).

Original entry on oeis.org

1, 4, 1, 12, 12, 1, 24, 96, 24, 1, 24, 600, 360, 40, 1, 0, 3024, 4200, 960, 60, 1, 0, 12096, 40824, 17640, 2100, 84, 1, 0, 36288, 338688, 270144, 55440, 4032, 112, 1, 0, 72576, 2407104, 3580416, 1212624, 144144, 7056, 144, 1, 0, 72576, 14515200, 41791680

Offset: 1

Wolfdieter Lang

Previous name was: A triangle of numbers related to triangle A049326.
a(n,1) = A008279(4,n-1). a(n,m) =: S1(-4; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A011801(n,m). The monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m, E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

			E.g., row polynomial E(3,x) = 12*x + 12*x^2 + x^3.
Triangle starts:
   1;
   4,   1;
  12,  12,   1;
  24,  96,  24,   1;
  24, 600, 360,  40,   1;

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Peter Luschny, The Bell transform

Cf. A049326.

Row sums give A049427.

rows = 10;
a[n_, m_] := BellY[n, m, Table[k! Binomial[4, k], {k, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

# uses[bell_matrix from A264428]
# Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.
bell_matrix(lambda n: factorial(n)*binomial(2, n), 8) # Peter Luschny, Jan 16 2016

a(n, m) = n!*A049326(n, m)/(m!*5^(n-m));
a(n, m) = (5*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1;
a(n, m) = 0, nE.g.f. for m-th column: (((-1+(1+x)^5)/5)^m)/m!.

New name from Peter Luschny, Jan 16 2016

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A157404 A partition product of Stirling_2 type [parameter k = 4] with biggest-part statistic (triangle read by rows).

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A144285 Lower triangular array called S2hat(-4) related to partition number array A144284.

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A049424 Triangle read by rows, the Bell transform of n!*binomial(4,n) (without column 0).

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