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

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

Author

Wolfdieter Lang, Nov 13 2007

Keywords

• nonn
• easy
• tabf

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

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

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

Formula

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).