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

%I A270913 #33 Apr 20 2023 11:50:46
%S A270913 1,1,3,13,51,206,855,3585,15155,64525,276278,1188353,5130999,22226049,
%T A270913 96544003,420368858,1834203955,8018057345,35107961175,153950675585,
%U A270913 675978772326,2971700764941,13078268135683,57613905606273,254038914924791,1121081799217231
%N A270913 Coefficient of x^n in Product_{k>=1} (1+x^k)^n.
%C A270913 From _Peter Bala_, Apr 18 2023: (Start)
%C A270913 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.
%C A270913 Conjecture: the supercongruence a(p) == p + 1 (mod p^2) holds for all primes p. (End)
%H A270913 Vaclav Kotesovec, <a href="/A270913/b270913.txt">Table of n, a(n) for n = 0..1000</a>
%F A270913 a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007... and c = A327280 = 0.260542233142438469433860832160...
%p A270913 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p A270913      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A270913     end:
%p A270913 g:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, b(n),
%p A270913        (q-> add(g(j, q)*g(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A270913     end:
%p A270913 a:= n-> g(n$2):
%p A270913 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jan 31 2021
%t A270913 Table[SeriesCoefficient[Product[(1+x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
%t A270913 Table[SeriesCoefficient[QPochhammer[-1, x]^n, {x, 0, n}]/2^n, {n, 0, 25}]
%t A270913 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 *)
%o A270913 (PARI) {a(n)=polcoeff(prod(k=1, n, (1 + x^k +x*O(x^n))^n), n)}
%o A270913 for(n=0, 20, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Aug 26 2019
%Y A270913 Cf. A000009, A008485, A255526, A270914, A270917, A270919, A324595, A362408.
%K A270913 nonn
%O A270913 0,3
%A A270913 _Vaclav Kotesovec_, Mar 25 2016