A270913 - OEIS

%I #33 Apr 20 2023 11:50:46

%S 1,1,3,13,51,206,855,3585,15155,64525,276278,1188353,5130999,22226049,

%T 96544003,420368858,1834203955,8018057345,35107961175,153950675585,

%U 675978772326,2971700764941,13078268135683,57613905606273,254038914924791,1121081799217231

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

%C From _Peter Bala_, Apr 18 2023: (Start)

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

%H Vaclav Kotesovec, <a href="/A270913/b270913.txt">Table of n, a(n) for n = 0..1000</a>

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

%p b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(

%p      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)

%p     end:

%p g:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, b(n),

%p        (q-> add(g(j, q)*g(n-j, k-q), j=0..n))(iquo(k, 2))))

%p     end:

%p a:= n-> g(n$2):

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Jan 31 2021

%t Table[SeriesCoefficient[Product[(1+x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]

%t Table[SeriesCoefficient[QPochhammer[-1, x]^n, {x, 0, n}]/2^n, {n, 0, 25}]

%t 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 (PARI) {a(n)=polcoeff(prod(k=1, n, (1 + x^k +x*O(x^n))^n), n)}

%o for(n=0, 20, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Aug 26 2019

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

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Mar 25 2016