A270913 Coefficient of x^n in Product_{k>=1} (1+x^k)^n.

1, 1, 3, 13, 51, 206, 855, 3585, 15155, 64525, 276278, 1188353, 5130999, 22226049, 96544003, 420368858, 1834203955, 8018057345, 35107961175, 153950675585, 675978772326, 2971700764941, 13078268135683, 57613905606273, 254038914924791, 1121081799217231

Offset: 0

Author

Vaclav Kotesovec, Mar 25 2016

Keywords

• nonn

Comments

From Peter Bala, Apr 18 2023: (Start)

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.

Conjecture: the supercongruence a(p) == p + 1 (mod p^2) holds for all primes p. (End)

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000009, A008485, A255526, A270914, A270917, A270919, A324595, A362408.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
g:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, b(n),
       (q-> add(g(j, q)*g(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> g(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 31 2021

Mathematica

Table[SeriesCoefficient[Product[(1+x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[QPochhammer[-1, x]^n, {x, 0, n}]/2^n, {n, 0, 25}]
Table[SeriesCoefficient[Exp[n*Sum[(-1)^j*x^j/(j*(x^j - 1)), {j, 1, n}]], {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, May 19 2018 *)

PARI

{a(n)=polcoeff(prod(k=1, n, (1 + x^k +x*O(x^n))^n), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 26 2019

Formula

a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007... and c = A327280 = 0.260542233142438469433860832160...