A008485 - OEIS

%I #59 Apr 20 2023 11:50:42

%S 1,1,5,22,105,506,2492,12405,62337,315445,1605340,8207563,42124380,

%T 216903064,1119974875,5796944357,30068145905,156250892610,

%U 813310723925,4239676354650,22130265931900,115654632452535,605081974091875,3168828466966388,16610409114771900

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

%C Number of partitions of n into parts of n kinds. - _Vladeta Jovovic_, Sep 08 2002

%C Main diagonal of A144064. - _Omar E. Pol_, Jun 27 2012

%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 >= 3. Cf. A270913. (End)

%H Alois P. Heinz, <a href="/A008485/b008485.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{pi} Product_{i=1..n} binomial(k_i+n-1, k_i) where pi runs through all nonnegative solutions of k_1+2*k_2+...+n*k_n=n. a(n) = b(n, n) where b(n, m)= m/n*Sum_{i=1..n} sigma(i)*b(n-i, m) is recurrence for number of partitions of n into parts of m kinds. - _Vladeta Jovovic_, Sep 08 2002

%F Equals the logarithmic derivative of A109085, the g.f. of which is (1/x)*Series_Reversion(x*eta(x)). - _Paul D. Hanna_, Apr 05 2012

%F Let G(x) = exp( Sum_{n>=1} a(n)*x^n/n ), then G(x) = 1/Product_{n>=1} (1-x^n*G(x)^n) is the g.f. of A109085. - _Paul D. Hanna_, Apr 05 2012

%F a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.352701333486642687772415814165..., c = A327279 = 0.26801521271073331568695383828... . - _Vaclav Kotesovec_, Sep 10 2014

%p with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:= n-> etr(j->n)(n): seq(a(n), n=0..30); # _Alois P. Heinz_, Sep 09 2008

%t a[n_] := SeriesCoefficient[ Product[1/(1-x^k)^n, {k, 1, n}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 1, 24}] (* _Jean-François Alcover_, Feb 24 2015 *)

%t Table[SeriesCoefficient[1/QPochhammer[x, x]^n, {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 25 2016 *)

%t Table[SeriesCoefficient[Exp[n*Sum[x^j/(j*(1-x^j)), {j, 1, n}]], {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, May 19 2018 *)

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

%o (PARI) {a(n)=n*polcoeff(log(1/x*serreverse(x*eta(x+x*O(x^n)))), n)} /* _Paul D. Hanna_, Apr 05 2012 */

%Y Cf. A000041, A000712, A000716, A023003-A023021, A005758, A006922.

%Y Cf. A109085, A192435, A252782, A255526, A255672, A270913, A270915, A270919.

%K nonn

%O 0,3

%A T. Forbes (anthony.d.forbes(AT)googlemail.com)

%E a(0)=1 prepended by _Alois P. Heinz_, Mar 30 2015