A049404 -id:A049404 - cp's OEIS frontend

A049425 Row sums of triangle A049404.

1, 1, 3, 9, 33, 141, 651, 3333, 18369, 108153, 678771, 4495041, 31324833, 228803589, 1744475643, 13852095741, 114235118721, 976176336753, 8627940414819, 78726234866553, 740440277799201, 7168107030092541, 71331617341611243, 728811735008913909, 7637128289949856833, 81995144342947130601

Offset: 0

Wolfdieter Lang

Seiichi Manyama, Table of n, a(n) for n = 0..616 (terms 0..200 from Vincenzo Librandi)
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, J. Integer Seqs., Vol. 20 (2017), #17.8.2.

Column k=2 of A293991.

Cf. A004212.

Table[n!*SeriesCoefficient[E^(x+x^2+(x^3)/3),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)

/* for b(n) = a(n+1) */
b(n) := sum((n!/k!)*sum(binomial(k,i)*binomial(k-i+2,n-2*i-k)/3^i,i,0,k),k,0,n);
makelist(b(n),n,0,24);  /* Emanuele Munarini, Oct 20 2014 */

x='x+O('x^66); Vec(serlaplace(exp(x+x^2+(x^3)/3))) \\ Joerg Arndt, May 04 2013

E.g.f.: exp(x+x^2+(x^3)/3).
a(n) = n! * sum(k=0..n, sum(j=0..k, binomial(3*j,n) * (-1)^(k-j)/(3^k * (k-j)!*j!))). [Vladimir Kruchinin, Feb 07 2011]
Conjecture: -a(n) +a(n-1) +(2*n-2)*a(n-2) + (2-3*n+n^2)*a(n-3)=0. - R. J. Mathar, Nov 14 2011
a(n) ~ exp(n^(2/3)+n^(1/3)/3-2*n/3-2/9)*n^(2*n/3)/sqrt(3)*(1+59/(162*n^(1/3))). - Vaclav Kotesovec, Oct 08 2012
From Emanuele Munarini, Oct 20 2014: (Start)
Recurrence: a(n+3) = a(n+2)+2*(n+2)*a(n+1)+(n+2)*(n+1)*a(n).
It derives from the differential equation for the e.g.f.: A'(x) = (1+2*x+x^2)*A(x).
So, the above conjecture is true.
b(n) = a(n+1) = sum((n!/k!)*sum(bin(k,i)*bin(k-i+2,n-2*i-k)/3^i,i=0..k),k=0..n).
E.g.f. for b(n) = a(n+1): (1+t)^2*exp(t+t^2+t^3/3).
(End)
a(n) = Sum_{k=0..n} Stirling1(n,k) * A004212(k). - Seiichi Manyama, Jan 31 2024
a(n) = (1/exp(1/3)) * n! * Sum_{k>=0} binomial(3*k,n)/(3^k * k!). - Seiichi Manyama, Jan 18 2025

A144358 Partition number array, called M31(-2), related to A049404(n,m) = S1(-2;n,m) (generalized Stirling triangle).

Original entry on oeis.org

1, 2, 1, 2, 6, 1, 0, 8, 12, 12, 1, 0, 0, 40, 20, 60, 20, 1, 0, 0, 0, 40, 0, 240, 120, 40, 180, 30, 1, 0, 0, 0, 0, 0, 0, 280, 840, 0, 840, 840, 70, 420, 42, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2240, 0, 0, 1120, 6720, 1680, 0, 2240, 3360, 112, 840, 56, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2240, 0, 0

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(-2;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=2) in the family M31(-K) of partition number arrays.
If M31(-2;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-2) := A049404.

			[1]; [2,1]; [2,6,1]; [0,8,12,12,1]; [0,0,40,20,60,20,1]; ...
a(4,3) = 12 = 3*S1(-2;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. A049425 (row sums).

Cf. A144357 (M31(-1) array), A144877 (M31(-3) array).

a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(S1(-2;j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S1(-2;j,1)^e(n,k,j),j=1..n) with S1(-2;n,1)|= A008279(2,n-1)= [1,2,2,0,...], 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.

A157392 A partition product of Stirling_1 type [parameter k = 2] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 1, 24, 8, 0, 1, 80, 60, 0, 0, 1, 330, 320, 0, 0, 0, 1, 1302, 2030, 0, 0, 0, 0, 1, 5936, 12432, 0, 0, 0, 0, 0, 1, 26784, 81368, 0, 0, 0, 0, 0, 0, 1, 133650, 545120, 0, 0, 0, 0, 0, 0, 0, 1, 669350, 3825690

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 = 2,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144358.
Same partition product with length statistic is A049404.
Diagonal a(A000217(n)) = falling_factorial(2,n-1), row in A008279
Row sum is A049425.

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+4).

A265604 Triangle read by rows: The inverse Bell transform of the quartic factorial numbers (A007696).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, -2, 3, 1, 0, 10, -5, 6, 1, 0, -80, 30, -5, 10, 1, 0, 880, -290, 45, 5, 15, 1, 0, -12320, 3780, -560, 35, 35, 21, 1, 0, 209440, -61460, 8820, -735, 0, 98, 28, 1, 0, -4188800, 1192800, -167300, 14700, -735, 0, 210, 36, 1

Offset: 0

Peter Luschny, Dec 30 2015

			[ 1]
[ 0,      1]
[ 0,      1,      1]
[ 0,     -2,      3,      1]
[ 0,     10,     -5,      6,      1]
[ 0,    -80,     30,     -5,     10,      1]
[ 0,    880,   -290,     45,      5,     15,      1]

Peter Luschny, The Bell transform
Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.

Cf. A007696, A264428, A264429.

Inverse Bell transforms of other multifactorials are: A048993, A049404, A049410, A075497, A075499, A075498, A119275, A122848, A265605.

# uses[bell_transform from A264428]
def inverse_bell_matrix(generator, dim):
    G = [generator(k) for k in srange(dim)]
    row = lambda n: bell_transform(n, G)
    M = matrix(ZZ, [row(n)+[0]*(dim-n-1) for n in srange(dim)]).inverse()
    return matrix(ZZ, dim, lambda n,k: (-1)^(n-k)*M[n,k])
multifact_4_1 = lambda n: prod(4*k + 1 for k in (0..n-1))
print(inverse_bell_matrix(multifact_4_1, 8))

A265605 Triangle read by rows: The inverse Bell transform of the triple factorial numbers (A007559).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, -1, 3, 1, 0, 3, -1, 6, 1, 0, -15, 5, 5, 10, 1, 0, 105, -35, 0, 25, 15, 1, 0, -945, 315, -35, 0, 70, 21, 1, 0, 10395, -3465, 490, -35, 70, 154, 28, 1, 0, -135135, 45045, -6895, 630, -105, 378, 294, 36, 1

Offset: 0

Peter Luschny, Dec 30 2015

			[ 1]
[ 0,    1]
[ 0,    1,    1]
[ 0,   -1,    3,    1]
[ 0,    3,   -1,    6,    1]
[ 0,  -15,    5,    5,   10,    1]
[ 0,  105,  -35,    0,   25,   15,    1]
[ 0, -945,  315,  -35,    0,   70,   21,    1]

Peter Luschny, The Bell transform
Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.

Cf. A007559, A264428, A264429.

Inverse Bell transforms of other multifactorials are: A048993, A049404, A049410, A075497, A075499, A075498, A119275, A122848, A265604.

# uses[bell_transform from A264428]
def inverse_bell_matrix(generator, dim):
    G = [generator(k) for k in srange(dim)]
    row = lambda n: bell_transform(n, G)
    M = matrix(ZZ, [row(n)+[0]*(dim-n-1) for n in srange(dim)]).inverse()
    return matrix(ZZ, dim, lambda n,k: (-1)^(n-k)*M[n,k])
multifact_3_1 = lambda n: prod(3*k + 1 for k in (0..n-1))
print(inverse_bell_matrix(multifact_3_1, 8))

A049324 A convolution triangle of numbers generalizing Pascal's triangle A007318.

Original entry on oeis.org

1, 3, 1, 3, 6, 1, 0, 15, 9, 1, 0, 18, 36, 12, 1, 0, 9, 81, 66, 15, 1, 0, 0, 108, 216, 105, 18, 1, 0, 0, 81, 459, 450, 153, 21, 1, 0, 0, 27, 648, 1305, 810, 210, 24, 1, 0, 0, 0, 594, 2673, 2970, 1323, 276, 27, 1, 0, 0, 0, 324, 3915, 7938

Offset: 1

Wolfdieter Lang

			{1}; {3,1}; {3,6,1}; {0,15,9,1}; {0,18,36,12,1}; ...

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

a(n, m) := s1(-2, n, m), a member of a sequence of triangles including s1(0, n, m)= A023531(n, m) (unit matrix) and s1(2, n, m)=A007318(n-1, m-1) (Pascal's triangle). s1(-1, n, m)= A030528.

Cf. A049348, A049404.

a(n, m) = 3*(3*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, nA033842(2, m)).

A331816 Irregular triangle (read by rows) of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..2n} T(n,k) x^k = (-1)^n * e^(x^3/3) * (((d/dx)^n) e^(-x^3/3)) for n >= 0 and 0 <= k <= 2*n.

Original entry on oeis.org

1, 0, 0, 1, 0, -2, 0, 0, 1, 2, 0, 0, -6, 0, 0, 1, 0, 0, 20, 0, 0, -12, 0, 0, 1, 0, -40, 0, 0, 80, 0, 0, -20, 0, 0, 1, 40, 0, 0, -360, 0, 0, 220, 0, 0, -30, 0, 0, 1, 0, 0, 1120, 0, 0, -1680, 0, 0, 490, 0, 0, -42, 0, 0, 1, 0, -2240, 0, 0, 9520, 0, 0, -5600, 0, 0, 952, 0, 0, -56, 0, 0, 1

Offset: 0

Werner Schulte, Jan 27 2020

Let r(s;n,x) = Sum_{k=0..s*n} A(s;n,k)*x^k = (-1)^n * e^(x^(s+1)/(s+1)) * (((d/dx)^n) e^(-x^(s+1)/(s+1))) for n >= 0 and x complex and some fixed integer s >= 1. Special cases: A(1;n,k) = A066325(n,k) and A(2;n,k) is this triangle. Formula: A(s;n,k) = (Sum_{i=0..floor(k/(s+1))} (-1)^i * binomial((n+k) /(s+1),i) * binomial(n+k-(s+1)*i,n)) * (-1)^(n-(n+k)/(s+1)) * (n!) / ((s+1)^((n+k)/(s+1)) * (((n+k)/(s+1))!)) if (n+k) mod (s+1) = 0 else 0 with n >= 0 and 0 <= k <= s*n.
  Recurrence: (1) A(s;n,k) = A(s;n-1,k-s) - (k+1) * A(s;n-1,k+1),
  (2) r(s;n,x) = x^s * r(s;n-1,x) - ((d/dx) r(s;n-1,x)) for n > 0 with initial values A(s;0,0) = 1 = r(s;0,x) and A(s;n,k) = 0 if k < 0 or k > s*n or (n+k) mod (s+1) > 0;
  E.g.f.: Sum_{n>=0} r(s;n,x)*t^n/(n!) = e^((x^(s+1)-(x-t)^(s+1))/(s+1)).
This generalization is result of a long and intensive discussion with Wolfdieter Lang. For more information see A091752.

			The irregular triangle T(n,k) starts:
n\k:  0     1    2    3    4     5   6     7   8   9  10   . . .      16
========================================================================
0  :  1
1  :  0     0    1
2  :  0    -2    0    0    1
3  :  2     0    0   -6    0     0   1
4  :  0     0   20    0    0   -12   0     0   1
5  :  0   -40    0    0   80     0   0   -20   0   0   1
6  : 40     0    0 -360    0     0 220     0   0 -30   0   0 1
7  :  0     0 1120    0    0 -1680   0     0 490   0   0 -42 0   0 1
8  :  0 -2240    0    0 9520     0   0 -5600   0   0 952   0 0 -56 0 0 1
etc.

Cf. A049404, A066325, A091752.

Row sums are (-1)^n*A252284(n).

T(n,k) = (-1)^k * (n!) * (Sum_{i=0..floor(k/3)} (-1)^i * binomial((n+k) /3,i) * binomial(n+k-3*i,n)) / (3^((n+k)/3) * ((n+k)/3)!) if (n+k) mod 3 = 0 else 0 with n >= 0 and 0 <= k <= 2*n.
Recurrence: (1) T(n,k) = T(n-1,k-2) - (k+1) * T(n-1,k+1),
  (2) T(n,k) = T(n-1,k-2) - 2*(n-1)*T(n-2,k-1) + (n-1)*(n-2)*T(n-3,k),
  (3) k*T(n,k) = 2*n*T(n-1,k-2) - n*(n-1)*T(n-2,k-1),
  (4) p(n,x) = x^2 * p(n-1,x) - (d/dx) p(n-1,x),
  (5) p(n,x) = x^2*p(n-1,x) - 2*(n-1)*x*p(n-2,x) + (n-1)*(n-2)*p(n-3,x),
  (6) (d/dx) p(n,x) = 2*n*x*p(n-1,x) - n*(n-1)*p(n-2,x) for n > 0 with initial values T(0,0) = 1 = p(0,x) and T(n,k) = 0 if k < 0 or k > 2*n or (n+k) mod 3 > 0.
T(n,2*n) = 1 for n >= 0.
T(3*n,0) = -T(3*n-1,1) = 2*T(3*n-2,2) = ((3*n)!)/(3^n * (n!)) for n > 0.
The polynomials p(n,x) satisfy for n >= 0 and x complex the differential equation: 0 = (((d/dx)^3) p(n,x)) - 2*x^2*(((d/dx)^2) p(n,x)) + (x^4 + 2*(n-1)*x) * ((d/dx) p(n,x)) - (2*n*x^3-(n+3)*n) * p(n,x).
E.g.f.: Sum_{n>=0} p(n,x)*t^n/(n!) = e^((x^3-(x-t)^3)/3).
((d/dx)^m) p(n,x) = Sum_{i=0..m} (-1)^i * binomial(m,i) * p(m-i,-x) * p(n+i,x) for m,n >= 0 and x complex.
T(3*n-k,k) = A091752(n+1,k+2) for 0 <= k <= 2*n.
(-1)^(n-k) * T(n,3*k-n) = A049404(n,k) for n > 0 and (n+2)/3 <= k <= n.

cp's OEIS Frontend

A049425 Row sums of triangle A049404.

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A144358 Partition number array, called M31(-2), related to A049404(n,m) = S1(-2;n,m) (generalized Stirling triangle).

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A157392 A partition product of Stirling_1 type [parameter k = 2] with biggest-part statistic (triangle read by rows).

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A265604 Triangle read by rows: The inverse Bell transform of the quartic factorial numbers (A007696).

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Sage

A265605 Triangle read by rows: The inverse Bell transform of the triple factorial numbers (A007559).

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Sage

A049324 A convolution triangle of numbers generalizing Pascal's triangle A007318.

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A331816 Irregular triangle (read by rows) of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..2*n} T(n,k) * x^k = (-1)^n * e^(x^3/3) * (((d/dx)^n) e^(-x^3/3)) for n >= 0 and 0 <= k <= 2*n.

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A331816 Irregular triangle (read by rows) of coefficients T(n,k) of polynomials p(n,x) = Sum_{k=0..2n} T(n,k) x^k = (-1)^n * e^(x^3/3) * (((d/dx)^n) e^(-x^3/3)) for n >= 0 and 0 <= k <= 2*n.