A179380 Triangle T(n,k) read by rows: product of A074664(a_i) over all parts a_i of the k-th partition of n listed in Abramowitz and Stegun order, 1 <= k <= A000041(n).

1, 1, 1, 1, 2, 1, 1, 6, 2, 1, 1, 1, 22, 6, 2, 2, 1, 1, 1, 92, 22, 6, 4, 6, 2, 1, 2, 1, 1, 1, 426, 92, 22, 12, 22, 6, 4, 2, 6, 2, 1, 2, 1, 1, 1, 2146, 426, 92, 44, 36, 92, 22, 12, 6, 4, 22, 6, 4, 2, 1, 6, 2, 1, 2, 1, 1, 1, 11624, 2146, 426, 184, 132, 426, 92, 44, 36, 22, 12, 8, 92, 22, 12

Offset

0,5

Comments

Row n has A000041(n) elements, sorted in Abramowitz-Stegun order (A036036).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)

Formula

A048996(n,k)* T(n,k) = A179313(n,k).

Sum_{k=1.. A000041(n)} T(n,k) = A179379(n).

T(n,1) = A074664(n).

Example

T(6,4) refers to the 4th partition of 6, 3+3. T(6,4)=A074664(3)*A074664(3)=2*2.
T(7,3) refers to the 3rd partition of 7, 2+5. T(7,3)=A074664(2)*A074664(5)=1*22.
The triangle starts:
  0 | 1;
  1 | 1;
  2 | 1,1;
  3 | 2,1,1;
  4 | 6,2,1,1,1;
  5 | 22,6,2,2,1,1,1;
  6 | 92,22,6,4,6,2,1,2,1,1,1;
  7 | 426,92,22,12,22,6,4,2,6,2,1,2,1,1,1;
  ...

Prog

(PARI)
Q(n)=Vec(1 - 1 / serlaplace( exp( exp( x + O(x*x^n)) - 1)))
C(sig, q)=prod(k=1, #sig, q[sig[k]])
Row(n)=my(q=Q(n)); [C(Vec(p), q) | p<-partitions(n)]
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 03 2025

Crossrefs

Cf. A000041 (row lengths), A036036, A048996, A074664, A179313, A179379 (row sums).

Sequence in context: A125731 A123361 A265315 * A107106 A178249 A114423

Adjacent sequences: A179377 A179378 A179379 * A179381 A179382 A179383

Keyword

nonn,tabf

Author

Alford Arnold, Jul 12 2010

Extensions

Edited and extended by R. J. Mathar, Jul 16 2010

a(0)=1 prepended by Andrew Howroyd, Oct 03 2025

Status

approved