1,2

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.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

W. Lang, First 10 rows and more.

a(n,k)= product(S2(3,j,1)^e(n,k,j),j=1..n) 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).

[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.

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

Sequence in context: A080924 A128042 A108083 this_sequence A134146 A085569 A072479

Adjacent sequences: A134142 A134143 A134144 this_sequence A134146 A134147 A134148

nonn,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Nov 13 2007