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

0,10

Links

Table of n, a(n) for n=0..61.

Eric Weisstein's World of Mathematics, Pyramidal Number

Index entries for sequences related to partitions

Index to sequences related to pyramidal numbers

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.

Example

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

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

Prog

(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

Crossrefs

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

Sequence in context: A306680 A083312 A032435 * A117502 A212630 A030360

Adjacent sequences: A337795 A337796 A337797 * A337799 A337800 A337801

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 22 2020

Extensions

More terms from Robert Israel, Sep 23 2020

Status

approved