A049385 -id:A049385 - cp's OEIS frontend

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.

1, 20, 1, 780, 60, 1, 45240, 4320, 120, 1, 3483480, 382200, 13800, 200, 1, 334414080, 40556880, 1734600, 33600, 300, 1, 38457619200, 5039012160, 243505080, 5699400, 69300, 420, 1

Offset: 1

Udita Katugampola, Mar 23 2013

			1;
20,1;
780,60,1;
45240,4320,120,1;
3483480,382200,13800,200,1;
334414080,40556880,1734600,33600,300,1;
38457619200,5039012160,243505080,5699400,69300,420,1;
5153320972800,718724260800,38155703040,1024322880,15262800,127680,560,1;

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

b[0]:=f(x):
for j from 1 to 10 do
b[j]:=simplify(x^20*diff(b[j-1],x$1);
end do;

A157396 A partition product of Stirling_2 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 6, 1, 18, 66, 1, 144, 264, 1056, 1, 600, 4620, 5280, 22176, 1, 4950, 68640, 110880, 133056, 576576, 1, 26586, 639870, 3141600, 3259872, 4036032, 17873856, 1, 234528, 10759056, 69263040, 105557760, 113008896, 142990848

Offset: 1

Peter Luschny, Mar 09 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -6,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A134278.
Same partition product with length statistic is A049385.
Diagonal a(A000217) = A008548.
Row sum is A049412.

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

Cf. A157397, A157398, 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}(-5*j - 1).

Offset corrected by Peter Luschny, Mar 14 2009

A049374 A triangle of numbers related to triangle A030527.

Original entry on oeis.org

1, 6, 1, 42, 18, 1, 336, 276, 36, 1, 3024, 4200, 960, 60, 1, 30240, 66024, 23400, 2460, 90, 1, 332640, 1086624, 557424, 87360, 5250, 126, 1, 3991680, 18805248, 13349952, 2916144, 255360, 9912, 168, 1, 51891840, 342486144, 325854144, 95001984

Offset: 1

Wolfdieter Lang

a(n,1) = A001725(n+4). a(n,m)=: S1p(6; 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), S1p(4; n,m) = A049352, S1p(5; n,m) = A049353(n,m).
Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A049385(n,m) =: S2(6; 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).
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+5 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007

			Triangle begins
       1;
       6,       1;
      42,      18,      1;
     336,     276,     36,     1;
    3024,    4200,    960,    60,    1;
   30240,   66024,  23400,  2460,   90,   1;
  332640, 1086624, 557424, 87360, 5250, 126, 1;
E.g., row polynomial E(3,x) = 42*x + 18*x^2 + x^3.
a(4,2) = 276 = 4*(6*7) + 3*(6*6) 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*6*7)=42 colored versions, e.g., ((1c1),(2c1,3c6,4c3)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 6 colors, c1..c6, can be chosen and the vertex labeled 4 with j=2 can come in 7 colors, e.g., c1..c7. Therefore there are 4*((1)*(1*6*7))=168 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*6)*(1*6))=108 such forests, e.g., ((1c1,3c4)(2c1,4c6)) or ((1c1,3c5)(2c1,4c2)), etc. - _Wolfdieter Lang_, Oct 12 2007

Muniru A Asiru, Table of n, a(n) for n = 1..1275
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First ten rows.

Cf. A049402 (row sums), A134140 (alternating row sums).

Flat(List([1..10],n->Factorial(n)*List([1..n],k->Sum([1..k],j->(-1)^(k-j)*Binomial(k,j)*Binomial(n+5*j-1,5*j-1)/(5^k*Factorial(k)))))); # Muniru A Asiru, Jun 23 2018

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+5)!/120, 10); # Peter Luschny, Jan 28 2016

a[n_, k_] = n!*Sum[(-1)^(k-j)*Binomial[k, j]*Binomial[n + 5j - 1, 5j - 1]/(5^k*k!), {j, 1, k}] ;
Flatten[Table[a[n, k], {n, 1, 9}, {k, 1, n}] ][[1 ;; 40]]
(* Jean-François Alcover, Jun 01 2011, 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[(#+5)!/120&, 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+5*j-1,5*j-1),j,1,k))/(5^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */

a(n,k)=(n!*sum(j=1,k,(-1)^(k-j)*binomial(k,j)*binomial(n+5*j-1,5*j-1)))/(5^k*k!);
for(n=1,12,for(k=1,n,print1(a(n,k),", "));print()); /* print triangle */ /* Joerg Arndt, Apr 01 2011 */

a(n, m) = n!*A030527(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, n < m; a(n, 0) := 0; a(1, 1)=1. E.g.f. for m-th column: ((x*(5 - 10*x + 10*x^2 - 5*x^3 + x^4)/(5*(1-x)^5))^m)/m!.

a(n,k) = n!* Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*binomial(n+5*j-1,5*j-1) /(5^k*k!). - Vladimir Kruchinin, Apr 01 2011

A134280 Triangle of numbers obtained from the partition array A134279.

Original entry on oeis.org

1, 6, 1, 66, 6, 1, 1056, 102, 6, 1, 22176, 1452, 102, 6, 1, 576576, 32868, 1668, 102, 6, 1, 17873856, 779328, 35244, 1668, 102, 6, 1, 643458816, 23912064, 843480, 36540, 1668, 102, 6, 1, 26381811456, 812173824, 25416072, 857736, 36540, 1668, 102, 6, 1

Offset: 1

Wolfdieter Lang, Nov 13 2007

This triangle is named S2(6)'.

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

			[1]; [6,1]; [66,6,1]; [1056,102,6,1]; [22176,1452,102,6,1]; ...

W. Lang, First 10 rows and more.

Cf. A134275 (S2(5)').
Cf. A134281 (row sums).
Cf. A134282 (alternating row sums).

a(n,m)=sum(product(S2(6;j,1)^e(n,m,k,j),j=1..n),k=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,k,j) is the exponent of j in the k-th m part partition of n. S2(6;j,1) = A049385(n,1) = A008548(n) = (5*n-4)(!^5)(quintuple- or 5-factorials).

A134279 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(6)/M_3.

Original entry on oeis.org

1, 6, 1, 66, 6, 1, 1056, 66, 36, 6, 1, 22176, 1056, 396, 66, 36, 6, 1, 576576, 22176, 6336, 4356, 1056, 396, 216, 66, 36, 6, 1, 17873856, 576576, 133056, 69696, 22176, 6336, 4356, 2376, 1056, 396, 216, 66, 36, 6, 1, 643458816, 17873856, 3459456, 1463616

Offset: 1

Wolfdieter Lang, Nov 13 2007

Partition number array M_3(6) = A134278 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short M_3(6)/M_3.
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.

			[1]; [6,1]; [66,6,1]; [1056,66,36,6,1]; [22176,1056,396,66,36,6,1]; ...

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.

Row sums give A134281 (also of triangle A134280).

Cf. A134274 (M_3(5)/M_3 partition array).

a(n,k) = Product_{j=1..n} S2(6,j,1)^e(n,k,j) with S2(6,n,1) = A049385(n,1) = A008548(n) = (5*n-4)(!^5) (quintuple- or 5-factorials) 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) = A134278(n,k)/A036040(n,k) (division of partition arrays M_3(6) 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

A223171 Triangle S(n,k) by rows: coefficients of 5^((n-1)/2)(x^(1/5)d/dx)^n when n is odd, and of 5^(n/2)(x^(4/5)d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 5, 6, 5, 6, 60, 25, 66, 110, 25, 66, 990, 825, 125, 1056, 2640, 1200, 125, 1056, 21120, 26400, 8000, 625, 22176, 73920, 50400, 10500, 625, 22176, 554400, 924000, 420000, 65625, 3125, 576576, 2402400, 2184000, 682500, 81250, 3125, 576576, 17297280

Offset: 0

Udita Katugampola, Mar 20 2013

			Triangle begins:
1;
1, 5;
6, 5;
6, 60, 25;
66, 110, 25;
66, 990, 825, 125;
1056, 2640, 1200, 125;
1056, 21120, 26400, 8000, 625;
22176, 73920, 50400, 10500, 625;
22176, 554400, 924000, 420000, 65625, 3125;
576576, 2402400, 2184000, 682500, 81250, 3125;
576576, 17297280, 36036000, 21840000, 5118750, 487500, 15625;
17873856, 89369280, 101556000, 42315000, 7556250, 581250, 15625;

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(5^((i+1)mod 2)*x^((3((i+1)mod 2)+1)/5)*(diff(a[i-1],x$1 )));
end do;

A223512 Triangle T(n,k) represents the coefficients of (x^10*d/dx)^n, where n=1,2,3,...;generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 10, 1, 190, 30, 1, 5320, 1060, 60, 1, 196840, 45600, 3400, 100, 1, 9054640, 2340040, 208800, 8300, 150, 1, 498005200, 140096880, 14241640, 690200, 17150, 210, 1, 31872332800, 9604302400, 1080045120, 60485040, 1856400, 31640, 280, 1, 2326680294400

Offset: 1

Udita Katugampola, Mar 23 2013

			1;
10,1;
190,30,1;
5320,1060,60,1;
196840,45600,3400,100,1;
9054640,2340040,208800,8300,150,1;
498005200,140096880,14241640,690200,17150,210,1;
31872332800,9604302400,1080045120,60485040,1856400,31640,280,1,2326680294400

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

b[0]:=g(x):
for j from 1 to 10 do
b[j]:=simplify(x^10*diff(b[j-1],x$1);
end do;

A223513 Triangle T(n,k) represents the coefficients of (x^11*d/dx)^n, where n=1,2,3,...

Original entry on oeis.org

1, 11, 1, 231, 33, 1, 7161, 1287, 66, 1, 293601, 61215, 4125, 110, 1, 14973651, 3476781, 279840, 10065, 165, 1, 913392711, 230534073, 21106701, 924000, 20790, 231, 1, 64850882481, 17511845967, 1771323246, 89482701, 2483250, 38346, 308, 1

Offset: 1

Udita Katugampola, Mar 23 2013

Generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

			1;
11,1;
231,33,1;
7161,1287,66,1;
293601,61215,4125,110,1;
14973651,3476781,279840,10065,165,1;
913392711,230534073,21106701,924000,20790,23,1;
64850882481,17511845967,1771323246,89482701,2483250,38346,308,1;

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

b[0]:=f(x):
for j from 1 to 10 do
b[j]:=simplify(x^11*diff(b[j-1],x$1);
end do;

cp's OEIS Frontend

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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A157396 A partition product of Stirling_2 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

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A049374 A triangle of numbers related to triangle A030527.

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A134280 Triangle of numbers obtained from the partition array A134279.

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A134279 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(6)/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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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.

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A223171 Triangle S(n,k) by rows: coefficients of 5^((n-1)/2)*(x^(1/5)*d/dx)^n when n is odd, and of 5^(n/2)*(x^(4/5)*d/dx)^n when n is even.

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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.

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.

A223171 Triangle S(n,k) by rows: coefficients of 5^((n-1)/2)(x^(1/5)d/dx)^n when n is odd, and of 5^(n/2)(x^(4/5)d/dx)^n when n is even.