A049427 -id:A049427 - cp's OEIS frontend

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

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).

A293991 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j=1..k+1} binomial(k,j-1)*x^j/j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 9, 10, 1, 1, 1, 5, 16, 33, 26, 1, 1, 1, 6, 25, 76, 141, 76, 1, 1, 1, 7, 36, 145, 436, 651, 232, 1, 1, 1, 8, 49, 246, 1025, 2776, 3333, 764, 1, 1, 1, 9, 64, 385, 2046, 8245, 19384, 18369, 2620, 1, 1, 1, 10, 81

Offset: 0

Seiichi Manyama, Oct 21 2017

			Square array A(n,k) begins:
   1,  1,   1,   1,    1, ...
   1,  1,   1,   1,    1, ...
   1,  2,   3,   4,    5, ...
   1,  4,   9,  16,   25, ...
   1, 10,  33,  76,  145, ...
   1, 26, 141, 436, 1025, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..5 give A000012, A000085, A049425, A049426, A049427, A049428.
Rows n=0-1 give A000012.
Main diagonal gives A294003.
Cf. A291709.

A[0, ] = 1; A[n, k_] := (n-1)!*Sum[Binomial[k, j-1]*A[n-j, k]/(n-j)!, {j, 1, Min[k+1, n]}];
Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2017 *)

E.g.f. of column k: exp(((1+x)^(k+1) - 1)/(k+1)).

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..min(k+1,n)} binomial(k,j-1)*A(n-j,k)/(n-j)! for n > 0.

A049402 Row sums of triangle A049374.

Original entry on oeis.org

1, 1, 7, 61, 649, 8245, 122215, 2069425, 39328465, 827226505, 19047582055, 475956135205, 12815133759385, 369605936607805, 11361372997850695, 370609338222772825, 12780705695068446625, 464412124831585889425, 17729002673226394402375, 709180766131239680070925

Offset: 0

Wolfdieter Lang

Alois P. Heinz, Table of n, a(n) for n = 0..415
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Column k=6 of A291709.

Cf. A049374, A049427.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+4)!/5!*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, j - 1]*(j + 4)!/5!*a[n - j], {j, 1, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

E.g.f. exp(p(x)) with p(x) := (1-(1-x)^5)/(5*(1-x)^5) (E.g.f. first column of A049374).
From Seiichi Manyama, Jan 18 2025: (Start)
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A005011(k).
a(n) = (1/exp(1/5)) * (-1)^n * n! * Sum_{k>=0} binomial(-5*k,n)/(5^k * k!). (End)

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

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.

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

A369756 Expansion of e.g.f. exp( (1 - (1+x)^5)/5 ).

Original entry on oeis.org

1, -1, -3, -1, 49, 255, -275, -13105, -83775, 170495, 8290045, 69257055, -111005135, -9684015745, -109196883795, -31470300625, 17728458119425, 276531029694975, 904537471692925, -44728487203650625, -1000823562359108175, -7110596979389965825

Offset: 0

Seiichi Manyama, Jan 31 2024

sign

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A001464, A252284, A369755.

Cf. A049427, A369753.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((1-(1+x)^5)/5)))

a(0) = 1; a(n) = -(n-1)! * Sum_{k=1..min(5,n)} binomial(4,k-1) * a(n-k)/(n-k)!.

a(n) = Sum_{k=0..n} 5^k * Stirling1(n,k) * Bell_k(-1/5), where Bell_n(x) is n-th Bell polynomial.

Showing 1-6 of 6 results.

cp's OEIS Frontend

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

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A293991 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j=1..k+1} binomial(k,j-1)*x^j/j).

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A049402 Row sums of triangle A049374.

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

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

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Sage

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A369756 Expansion of e.g.f. exp( (1 - (1+x)^5)/5 ).

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