login
A238962
Number of perfect partitions in graded colexicographic order.
2
1, 1, 2, 3, 4, 8, 13, 8, 20, 26, 44, 75, 16, 48, 76, 132, 176, 308, 541, 32, 112, 208, 252, 368, 604, 818, 1076, 1460, 2612, 4683, 64, 256, 544, 768, 976, 1888, 2316, 3172, 3408, 5740, 7880, 10404, 14300, 25988, 47293, 128, 576, 1376, 2208, 2568, 2496, 5536, 7968
OFFSET
0,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014, Table A.1 entry |P^T(s)|.
FORMULA
T(n,k) = A074206(A036035(n,k)). - Andrew Howroyd, Apr 25 2020
EXAMPLE
Triangle T(n,k) begins:
1;
1;
2, 3;
4, 8, 13;
8, 20, 26, 44, 75;
16, 48, 76, 132, 176, 308, 541;
32, 112, 208, 252, 368, 604, 818, 1076, 1460, 2612, 4683;
...
PROG
(PARI) \\ here b(n) is A074206.
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
b(n)={if(!n, 0, my(sig=factor(n)[, 2], m=vecsum(sig)); sum(k=0, m, prod(i=1, #sig, binomial(sig[i]+k-1, k-1))*sum(r=k, m, binomial(r, k)*(-1)^(r-k))))}
Row(n)={apply(s->b(N(s)), [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Aug 30 2020
CROSSREFS
Row sums are A035341.
Cf. A002033 in graded colexicographic order.
Sequence in context: A226947 A272615 A356188 * A238975 A098348 A131420
KEYWORD
nonn,tabf
AUTHOR
Sung-Hyuk Cha, Mar 07 2014
EXTENSIONS
Offset changed and terms a(44) and beyond from Andrew Howroyd, Apr 25 2020
STATUS
approved