A134273 -id:A134273 - cp's OEIS frontend

A134278 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(6).

1, 6, 1, 66, 18, 1, 1056, 264, 108, 36, 1, 22176, 5280, 3960, 660, 540, 60, 1, 576576, 133056, 95040, 43560, 15840, 23760, 3240, 1320, 1620, 90, 1, 17873856, 4036032, 2794176, 2439360, 465696, 665280, 304920, 249480, 36960, 83160, 22680, 2310

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(6), the k=6 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(6,n,m):=A049385(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 6-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.

			[1]; [6,1]; [66,18,1]; [1056,264,108,36,1]; [22176,5280,3960,660,540,60,1]; ...
There are a(4,3) = 108 = 3*6^2 unordered 2-forests with 4 vertices, composed of two 6-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 six versions from the 6-ary 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].
Wolfdieter Lang, First 10 rows and more.

Cf. A049412 (row sums, also of triangle A049385).

Cf. A134273 (M_3(5) partition array).

a(n,k) = n!*Product_{j=1..n} (S2(6,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(6,n,1) = A049385(n,1) = A008548(n) = (5*n-4)(!^5) (quintuple- or 5-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.

A157397 A partition product of Stirling_2 type [parameter k = -5] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 5, 1, 15, 45, 1, 105, 180, 585, 1, 425, 2700, 2925, 9945, 1, 3075, 34650, 52650, 59670, 208845, 1, 15855, 308700, 1248975, 1253070, 1461915, 5221125, 1, 123515, 4475520, 23689575, 33972120, 35085960, 41769000

Offset: 1

Peter Luschny, Mar 09 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -5,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A134273.
Same partition product with length statistic is A049029.
Diagonal a(A000217) = A007696.
Row sum is A049120.

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

Cf. A157396, 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}(-4*j - 1).

Offset corrected by Peter Luschny, Mar 14 2009

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).

cp's OEIS Frontend

A134278 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(6).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A157397 A partition product of Stirling_2 type [parameter k = -5] with biggest-part statistic (triangle read by rows).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula