A008485 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^n.

1, 1, 5, 22, 105, 506, 2492, 12405, 62337, 315445, 1605340, 8207563, 42124380, 216903064, 1119974875, 5796944357, 30068145905, 156250892610, 813310723925, 4239676354650, 22130265931900, 115654632452535, 605081974091875, 3168828466966388, 16610409114771900

Offset

0,3

Comments

Number of partitions of n into parts of n kinds. - Vladeta Jovovic, Sep 08 2002

Main diagonal of A144064. - Omar E. Pol, Jun 27 2012

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 >= 3. Cf. A270913. (End)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

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

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

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

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.352701333486642687772415814165..., c = A327279 = 0.26801521271073331568695383828... . - Vaclav Kotesovec, Sep 10 2014

Maple

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

Mathematica

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 *)
Table[SeriesCoefficient[1/QPochhammer[x, x]^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 25 2016 *)
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 *)

Prog

(PARI) {a(n)=polcoeff(prod(k=1, n, 1/(1-x^k +x*O(x^n))^n), n)}
(PARI) {a(n)=n*polcoeff(log(1/x*serreverse(x*eta(x+x*O(x^n)))), n)} /* Paul D. Hanna, Apr 05 2012 */

Crossrefs

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

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

Sequence in context: A017972 A363546 A296163 * A213684 A082297 A267241

Adjacent sequences: A008482 A008483 A008484 * A008486 A008487 A008488

Keyword

nonn

Author

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

Extensions

a(0)=1 prepended by Alois P. Heinz, Mar 30 2015

Status

approved