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(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.

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(4,j,1)^e(n,k,j),j=1..n) 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).

[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.

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

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

Sequence in context: A139051 A061692 A096206 this_sequence A134151 A119304 A114150

Adjacent sequences: A134147 A134148 A134149 this_sequence A134151 A134152 A134153

nonn,easy,tabf

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