A049029 -id:A049029 - cp's OEIS frontend

A134273 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5).

1, 5, 1, 45, 15, 1, 585, 180, 75, 30, 1, 9945, 2925, 2250, 450, 375, 50, 1, 208845, 59670, 43875, 20250, 8775, 13500, 1875, 900, 1125, 75, 1, 5221125, 1461915, 1044225, 921375, 208845, 307125, 141750, 118125, 20475, 47250, 13125, 1575, 2625, 105, 1

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(5), the k=5 member in the family of a generalization of the multinomial number arrays 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(5,n,m):=A049029(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 quintic (5-ary) 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.

			Triangle begins:
  [1];
  [51];
  [45,15,1];
  [585,180,75,30,1];
  [9945,2925,2250,450,375,50,1];
  ...

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].
Wolfdieter Lang, First 10 rows and more.

Cf. There are a(4, 3)=75=3*5^2 unordered 2-forest with 4 vertices, composed of two 5-ary increasing 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 five versions from the 5-ary structure.
Cf. A049120 (row sums also of triangle A049029).
Cf. A134149 (M_3(4) array).

a(n,k) = n!*Product_{j=1..n} (S2(5,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(5,n,1) = A049029(n,1) = A007696(n) = (4*n-3)(!^4) (quadruple- or 4-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.

A134274 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5)/M_3.

Original entry on oeis.org

1, 5, 1, 45, 5, 1, 585, 45, 25, 5, 1, 9945, 585, 225, 45, 25, 5, 1, 208845, 9945, 2925, 2025, 585, 225, 125, 45, 25, 5, 1, 5221125, 208845, 49725, 26325, 9945, 2925, 2025, 1125, 585, 225, 125, 45, 25, 5, 1, 151412625, 5221125, 1044225, 447525, 342225

Offset: 1

Wolfdieter Lang, Nov 13 2007

Partition number array M_3(5) = A134273 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short M_3(5)/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.

			Triangle begins:
  [1];
  [5,1];
  [45,5,1];
  [585,45,25,5,1];
  [9945,585,225,45,25,5,1];
  ...

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].
Wolfdieter Lang, First 10 rows and more.

Row sums A134276 (also of triangle A134275).

Cf. A134150 (M_3(4)/M_3 array).

a(n,k) = Product_{j=1..n} S2(5,j,1)^e(n,k,j) with S2(5,n,1) = A049029(n,1) = A007696(n) = (4*n-3)(!^4) (quadruple- or 4-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) = A134273(n,k)/A036040(n,k) (division of partition arrays M_3(5) by M_3).

A143169 Fourth column of triangle A000369: |S2(-3;n+4,4)|.

Original entry on oeis.org

1, 30, 825, 24150, 775845, 27478710, 1069801425, 45547251750, 2108878296525, 105616706545350, 5693005525232025, 328784072492625750, 20261087389388971125, 1327378299252353097750, 92142485069345244158625, 6756933615539839013031750, 522007423480304780922028125

Offset: 0

Wolfdieter Lang, Sep 15 2008

Also third column (m=3) of triangle A049029 (S2(5)).

Third column of A000369 is A143168, fifth one is A143170.

Also third column (m=3) of triangle A049029 (S2(5)). - Wolfdieter Lang, Nov 17 2008

a(n) = A000369(n+4,4) = |S2(-3;n+4,4)|, n >= 0.

E.g.f.: d^4/dx^4 ((1-(1-4*x)^(1/4))^4)/4! = (-1/2)*(-45*(1-4*x)^(1/2)+120*(1-4*x)^(1/4)-77)/(1-4*x)^(15/4).

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;

A371080 Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3.

Original entry on oeis.org

1, 0, 1, 0, 4, 1, 0, 28, 12, 1, 0, 280, 160, 24, 1, 0, 3640, 2520, 520, 40, 1, 0, 58240, 46480, 11880, 1280, 60, 1, 0, 1106560, 987840, 295960, 40040, 2660, 84, 1, 0, 24344320, 23826880, 8090880, 1296960, 109200, 4928, 112, 1

Offset: 0

Peter Luschny, Mar 12 2024

			Triangle starts:
[0] 1;
[1] 0,       1;
[2] 0,       4,      1;
[3] 0,      28,     12,      1;
[4] 0,     280,    160,     24,     1;
[5] 0,    3640,   2520,    520,    40,    1;
[6] 0,   58240,  46480,  11880,  1280,   60,  1;
[7] 0, 1106560, 987840, 295960, 40040, 2660, 84, 1;

Peter Luschny, The Bell transform.

Variant: A035469.
Cf. A264428, A371076, A371077.
Cf. A007559, A144346.
Cf. A035342, A049029, A271703.

a := n -> mul(select(k -> k mod 3 = 1, [seq(1..3*(n + 1))])): BellMatrix(a, 9);
# Alternative:
BellMatrix(n -> coeff(series((1/x)*hypergeom([1, 1/3], [], 3*x),x, 22), x, n), 9);
# Recurrence:
T := proc(n, k) option remember; if k = n then 1 elif k = 0 then 0 else
T(n - 1, k - 1) + (3*(n - 1) + k) * T(n - 1, k) fi end:
for n from 0 to 7 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Mar 13 2024

T(n, k) = sum(j=k, n, 3^(n-j)*abs(stirling(n, j, 1))*stirling(j, k, 2)); \\ Seiichi Manyama, Apr 19 2025

T(n, k) = BellMatrix([x^n] hypergeom2F0([1, 1/3], [], 3*x) / x).
T(n, k) = A371076(n, k) / k!.
From Werner Schulte, Mar 13 2024: (Start)
T(n, k) = (Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * Product_{j=0..n-1} (3*j + i)) / (k!).
T(n, k) = T(n-1, k-1) + (3*(n - 1) + k) * T(n-1, k) for 0 < k < n with initial values T(n, 0) = 0 for n > 0 and T(n, n) = 1 for n >= 0. (End)
From Seiichi Manyama, Apr 19 2025: (Start)
T(n,k) = Sum_{j=k..n} 3^(n-j) * |Stirling1(n,j)| * Stirling2(j,k).
E.g.f. of column k (with leading zeros): (1/(1 - 3*x)^(1/3) - 1)^k / k!. (End)

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;

A223514 Triangle T(n,k) represents the coefficients of (x^12*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, 12, 1, 276, 36, 1, 9384, 1536, 72, 1, 422280, 80040, 4920, 120, 1, 23647680, 4984560, 365400, 12000, 180, 1, 1584394560, 362597760, 30197160, 1205400, 24780, 252, 1, 123582775680, 30229617600, 2778370560, 127834560, 3237360, 45696, 336, 1, 1099867035520

Offset: 1

Udita Katugampola, Mar 23 2013

			1;
12,1;
276,36,1;
9384,1536,72,1;
422280,80040,4920,120,1;
23647680,4984560,365400,12000,180,1;
1584394560,362597760,30197160,1205400,24780,252,1;
123582775680,30229617600,2778370560,127834560,3237360,45696,336,1;
1099867035520,...

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^12*diff(b[j-1],x$1);
end do;

cp's OEIS Frontend

A134273 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5).

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A134274 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5)/M_3.

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A143169 Fourth column of triangle A000369: |S2(-3;n+4,4)|.

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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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Examples

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Programs

Maple

A371080 Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3.

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Formula

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

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

Views

Author

Keywords

Comments

Examples

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.