A035469 -id:A035469 - cp's OEIS frontend

A049352 A triangle of numbers related to triangle A030524.

1, 4, 1, 20, 12, 1, 120, 128, 24, 1, 840, 1400, 440, 40, 1, 6720, 16240, 7560, 1120, 60, 1, 60480, 201600, 129640, 27720, 2380, 84, 1, 604800, 2681280, 2275840, 656320, 80080, 4480, 112, 1, 6652800, 38142720, 41370560, 15402240, 2498160, 196560

Offset: 1

Wolfdieter Lang

a(n,1) = A001715(n+2). a(n,m)=: S1p(4; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries including S1p(1; n,m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n,m) (unsigned Lah numbers), S1p(3; n,m)= A046089(n,m).
The signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A035469(n,m) := S2(4; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+3 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of A001715. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			Triangle starts:
{1};
{4,1};
{20,12,1};
{120,128,24,1};
{840,1400,440,40,1};
...
E.g. Row polynomial E(3,x)= 20*x + 12*x^2 + x^3.
a(4,2)=128=4*(4*5)+3*(4*4) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*4*5)=20 colored versions, e.g. ((1c1),(2c1,3c4,4c3)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 4 colors, c1..c4, can be chosen and the vertex labeled 4 with j=2 can come in 5 colors, e.g. c1..c5. Therefore there are 4*((1)*(1*4*5))=80 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*4)*(1*4))=48 such forests, e.g. ((1c1,3c2)(2c1,4c4)) or ((1c1,3c3)(2c1,4c2)), etc. - _Wolfdieter Lang_, Oct 12 2007

Wolfdieter Lang, First ten rows.

Cf. A049377 (row sums).

Alternating row sums A134137.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+3)!/6, 10); # Peter Luschny, Jan 28 2016

a[n_, k_] := (n!* Sum[(-1)^(k-j)*Binomial[k, j]*Binomial[n+3*j-1, 3*j-1], {j, 1, k}])/(3^k*k!); Table[a[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 26 2013, after Vladimir Kruchinin *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[(# + 3)!/6&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

a(n,k):=(n!*sum((-1)^(k-j)*binomial(k,j)*binomial(n+3*j-1,3*j-1),j,1,k))/(3^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */

a(n, m) = n!*A030524(n, m)/(m!*3^(n-m)); a(n, m) = (3*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n

a(n,k) = (n!*sum(j=1..k, (-1)^(k-j)*binomial(k,j)*binomial(n+3*j-1,3*j-1)))/(3^k*k!). [Vladimir Kruchinin, Apr 01 2011]

A157398 A partition product of Stirling_2 type [parameter k = -4] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 4, 1, 12, 28, 1, 72, 112, 280, 1, 280, 1400, 1400, 3640, 1, 1740, 15120, 21000, 21840, 58240, 1, 8484, 126420, 401800, 382200, 407680, 1106560, 1, 57232, 1538208, 6370000, 8357440, 8153600, 8852480, 24344320, 1

Offset: 1

Peter Luschny, Mar 09 2009, Mar 14 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 A134149.
Same partition product with length statistic is A035469.
Diagonal a(A000217) = A007559.
Row sum is A049119.

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

Cf. A157396, A157397, A157399, A157400, A080510, A157401, A157402, A157403, A157404, 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}(-3*j - 1).

A134151 Triangle of numbers obtained from the partition array A134150.

Original entry on oeis.org

1, 4, 1, 28, 4, 1, 280, 44, 4, 1, 3640, 392, 44, 4, 1, 58240, 5544, 456, 44, 4, 1, 1106560, 80640, 5992, 456, 44, 4, 1, 24344320, 1519840, 88256, 6248, 456, 44, 4, 1, 608608000, 31420480, 1631392, 90048, 6248, 456, 44, 4, 1, 17041024000, 766525760, 33293120

Offset: 1

Wolfdieter Lang Nov 13 2007

This triangle is named S2(4)'.

In the same manner the unsigned Lah triangle A008297 is obtained from the partition array A130561.

			[1]; [4,1]; [28,4,1]; [280,44,4,1]; [3640,392,44,4,1];...

W. Lang, First 10 rows and more.

Cf. A134152 (row sums). A134272 (alternating row sums).

Cf. A134146 (S2(3)' triangle).

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;j,1)= A007559(j) = A035469(j,1) = (3*j-2)!!!.

A134149 A certain partition array in Abramowitz-Stegun (A-St) order.

Original entry on oeis.org

1, 4, 1, 28, 12, 1, 280, 112, 48, 24, 1, 3640, 1400, 1120, 280, 240, 40, 1, 58240, 21840, 16800, 7840, 4200, 6720, 960, 560, 720, 60, 1, 1106560, 407680, 305760, 274400, 76440, 117600, 54880, 47040, 9800, 23520, 6720, 980, 1680, 84, 1, 24344320

Offset: 1

Wolfdieter Lang, Nov 13 2007

For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_3(4), the k=4 member of a family of generalizations of the multinomial number array M_3 = M_3(1) = A036040.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
The S2(4,n,m) numbers (generalized Stirling2 numbers) are obtained by summing in row n all numbers with the same part number m. In the same manner the S2(n,m) (Stirling2) numbers A008277 are obtained from the partition array M_3= A036040.
a(n,k) enumerates unordered forests of increasing quaternary trees related to the k-th partition of n in the A-St order. The m-forest is composed of m such trees, with m the number of parts of the partition.

			[1]; [4,1]; [28,12,1]; [280,112,48,24,1]; [3640,1400,1120,280,240,40,1]; ...
a(4,3)=48 from the third (k=3) partition (2^2) of 4: 4!*((4/2!)^2)/2 = 48, because S2(4,2,1) = 4!!! = 4*1 = 4.
There are a(4,3) = 48 = 3*4^2 unordered 2-forests with 4 vertices, composed of two increasing quaternary (4-ary) trees, each with two vertices: there are 3 increasing labelings (1,2)(3,4); (1,3)(2,4); (1,4)(2,3) and each tree comes in four versions from the quaternary structure.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Cf. A134144 (M_3(3) array).

a(n,k) = n!*Product_{j=1..n} (S2(4,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(4,n,1) = A035469(n,1) = A007559(n) = (3*n-2)!!! (triple- or 3-factorials) 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. Exponents 0 can be omitted due to 0!=1.

A134150 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(4)/M_3.

Original entry on oeis.org

1, 4, 1, 28, 4, 1, 280, 28, 16, 4, 1, 3640, 280, 112, 28, 16, 4, 1, 58240, 3640, 1120, 784, 280, 112, 64, 28, 16, 4, 1, 1106560, 58240, 14560, 7840, 3640, 1120, 784, 448, 280, 112, 64, 28, 16, 4, 1, 24344320, 1106560, 232960, 101920, 78400, 58240, 14560, 7840

Offset: 1

Wolfdieter Lang, Nov 13 2007

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_3(4) = A134149 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short, M_3(4)/M_3.

			Triangle begins:
  [1];
  [4,1];
  [28,4,1];
  [280,28,16,4,1];
  [3640,280,112,28,16,4,1];
  ...
a(4,3)=16 from the third (k=3) partition (2^2) of 4: (4)^2 = 16, because S2(4,2,1) = 4!! = 4*1 = 4.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Wolfdieter Lang, First 10 rows and more.

Cf. A134145 (M_3(3)/M_3 array).

Cf. A134152 (row sums, also of triangle A134151).

a(n,k) = Product_{j=1..n} S2(4,j,1)^e(n,k,j) with S2(4,n,1) = A035469(n,1) = A007559(n) = (3*n-2)!!! and with 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.

a(n,k) = A134149(n,k)/A036040(n,k) (division of partition arrays M_3(4) by M_3).

A223169 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)(x^(1/3)d/dx)^n when n is odd, and of 3^(n/2)(x^(2/3)d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 3, 4, 3, 4, 24, 9, 28, 42, 9, 28, 252, 189, 27, 280, 630, 270, 27, 280, 3360, 3780, 1080, 81, 3640, 10920, 7020, 1404, 81, 3640, 54600, 81900, 35100, 5265, 243, 58240, 218400, 187200, 56160, 6480, 243, 58240, 1048320, 1965600

Offset: 0

Udita Katugampola, Mar 18 2013

			Triangle begins:
1;
1, 3;
4, 3;
4, 24, 9;
28, 42, 9;
28, 252, 189, 27;
280, 630, 270, 27;
280, 3360, 3780, 1080, 81;
3640, 10920, 7020, 1404, 81;
3640, 54600, 81900, 35100, 5265, 243,
58240, 218400, 187200, 56160, 6480, 243

U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.
U. N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229, 2014

Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

a[0]:= f(x):
for i from 1 to 13 do
a[i] := simplify(3^((i+1)mod 2)*x^(((i+1)mod 2+1)/3)*(diff(a[i-1],x$1 )));
end do;

A223170 Triangle S(n,k) by rows: coefficients of 4^((n-1)/2)(x^(1/4)d/dx)^n when n is odd, and of 4^(n/2)(x^(3/4)d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 4, 5, 4, 5, 40, 16, 45, 72, 16, 45, 540, 432, 64, 585, 1404, 624, 64, 585, 9360, 11232, 3328, 256, 9945, 31824, 21216, 4352, 256, 9945, 198900, 318240, 141440, 21760, 1024, 208845, 835380, 742560, 228480, 26880, 1024, 208845, 5012280, 10024560, 5940480, 1370880, 129024, 4096

Offset: 0

Udita Katugampola, Mar 20 2013

			Triangle begins:
1;
1, 4;
5, 4;
5, 40, 16;
45, 72, 16;
45, 540, 432, 64;
585, 1404, 624, 64;
585, 9360, 11232, 3328, 256;
9945, 31824, 21216, 4352, 256;
9945, 198900, 318240, 141440, 21760, 1024;
208845, 835380, 742560, 228480, 26880, 1024;
208845, 5012280, 10024560, 5940480, 1370880, 129024, 4096;

U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.
U. N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229, 2014

Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

a[0]:= f(x):
for i from 1 to 13 do
a[i] := simplify(4^((i+1)mod 2)*x^((2((i+1)mod 2)+1)/4)*(diff(a[i-1],x$1 )));
end do;

nmax = 12;
b[0] = Exp[x]; For[ i = 1 , i <= nmax , i++, b[i] = 4^Mod[i + 1, 2]*x^((2 Mod[i + 1, 2] + 1)/4)*D[b[i - 1], x]] // Simplify;
row[1] = {1}; row[n_] := List @@ Expand[b[n]/f[x]] /. x -> 1;
Table[row[n], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Feb 22 2019, from Maple *)

Missing terms inserted by Jean-François Alcover, Feb 22 2019

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A223532 Triangle S(n,k) by rows: coefficients of 6^(n/2)*(x^(5/6)*d/dx)^n when n=0,2,4,6,...

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A223511 Triangle T(n,k) represents the coefficients of (x^9*d/dx)^n, where n=1,2,3,...;generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

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A223522 Triangle T(n,k) represents the coefficients of (x^20*d/dx)^n, where n=1,2,3,...; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

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A049352 A triangle of numbers related to triangle A030524.

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A157398 A partition product of Stirling_2 type [parameter k = -4] with biggest-part statistic (triangle read by rows).

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A134151 Triangle of numbers obtained from the partition array A134150.

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A134149 A certain partition array in Abramowitz-Stegun (A-St) order.

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A134150 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(4)/M_3.

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A223169 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)*(x^(1/3)*d/dx)^n when n is odd, and of 3^(n/2)*(x^(2/3)*d/dx)^n when n is even.

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A223532 Triangle S(n,k) by rows: coefficients of 6^(n/2)(x^(5/6)d/dx)^n when n=0,2,4,6,...

A223169 Triangle S(n,k) by rows: coefficients of 3^((n-1)/2)(x^(1/3)d/dx)^n when n is odd, and of 3^(n/2)(x^(2/3)d/dx)^n when n is even.

A223170 Triangle S(n,k) by rows: coefficients of 4^((n-1)/2)(x^(1/4)d/dx)^n when n is odd, and of 4^(n/2)(x^(3/4)d/dx)^n when n is even.