A337798 Number of partitions of the n-th n-gonal pyramidal number into distinct n-gonal pyramidal numbers.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 4, 5, 4, 5, 7, 11, 9, 4, 12, 12, 24, 23, 42, 59, 64, 58, 124, 206, 212, 168, 377, 539, 703, 873, 1122, 1505, 1943, 2724, 4100, 4513, 6090, 7138, 12079, 16584, 20240, 27162, 35874, 52622, 69817, 88059, 115628, 152756, 219538, 240200, 358733, 480674

Offset: 0

Author

Ilya Gutkovskiy, Sep 22 2020

Keywords

• nonn

Examples

			a(9) = 2 because the ninth 9-gonal pyramidal number is 885 and we have [885] and [420, 266, 155, 34, 10].
		

Links

• Eric Weisstein's World of Mathematics, Pyramidal Number
• Index entries for sequences related to partitions
• Index to sequences related to pyramidal numbers

Crossrefs

Cf. A006484, A288126, A298857, A337763, A337797, A337799.

Programs

Maple

p:= (n,k) ->  k * (k + 1) * (k * (n - 2) - n + 5) / 6:
f:= proc(n) local k, P;
  P:= mul(1+x^p(n,k),k=1..n);
  coeff(P,x,p(n,n));
end proc:
map(f, [$0..80]); # Robert Israel, Sep 23 2020

PARI

default(parisizemax, 2^31);
p(n,k) = k*(k + 1)*(k*(n-2) - n + 5)/6;
a(n) = my(f=1+x*O(x^p(n,n))); for(k=1, n, f*=1+x^p(n,k)); polcoeff(f, p(n,n)); \\ Jinyuan Wang, Dec 21 2021

Formula

a(n) = [x^p(n,n)] Product_{k=1..n} (1 + x^p(n,k)), where p(n,k) = k * (k + 1) * (k * (n - 2) - n + 5) / 6 is the k-th n-gonal pyramidal number.

Extensions

More terms from Robert Israel, Sep 23 2020