A134145 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(3)/M_3.

1, 3, 1, 15, 3, 1, 105, 15, 9, 3, 1, 945, 105, 45, 15, 9, 3, 1, 10395, 945, 315, 225, 105, 45, 27, 15, 9, 3, 1, 135135, 10395, 2835, 1575, 945, 315, 225, 135, 105, 45, 27, 15, 9, 3, 1, 2027025, 135135, 31185, 14175, 11025, 10395, 2835, 1575, 945, 675, 945, 315, 225

Offset: 1

Author

Wolfdieter Lang, Nov 13 2007

Keywords

• nonn
• easy

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(3) = A134144 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short M_3(3)/M_3.

Examples

			[1]; [3,1]; [15,3,1]; [105,15,9,3,1]; [945,105,45,15,9,3,1]; ...
a(4,3)=9 from the third (k=3) partition (2^2) of 4: (3)^2 = 9, because S2(3,2,1) = 3!! = 1*3 = 3.
		

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].
• W. Lang, First 10 rows and more.

Crossrefs

Cf. A134147 (row sums, also of triangle A134146).

Formula

a(n,k) = Product_{j=1..n} S2(3,j,1)^e(n,k,j) with S2(3,n,1) = A035342(n,1) = A001147(n) = (2*n-1)!! 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) = A134144(n,k)/A036040(n,k) (division of partition arrays M_3(3) by M_3).