A134152 -id:A134152 - cp's OEIS frontend

A134151 Triangle of numbers obtained from the partition array A134150.

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)!!!.

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).

A134272 Alternating row sums of triangle A134151.

Original entry on oeis.org

1, 3, 25, 239, 3289, 53111, 1031497, 22906903, 578734697, 16306214743, 508360540457, 17354157283287, 644314385719593, 25842490815894231, 1113789927340848937, 51332921853722842327, 2519491205735072674601

Offset: 1

Wolfdieter Lang, Nov 13 2007

Cf. A134152 (row sums of triangle A134151).

a(n) = Sum_{m=1..n} A134151(n,m)*(-1)^(m-1), n>=1.

cp's OEIS Frontend

A134151 Triangle of numbers obtained from the partition array A134150.

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A134272 Alternating row sums of triangle A134151.

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Author

Keywords

Crossrefs

Formula