A206227 - OEIS

%I #20 May 25 2015 06:29:02

%S 1,1,4,19,108,674,4494,31275,225132,1662894,12541802,96225037,

%T 748935563,5900502806,46976736513,377425326138,3056671009814,

%U 24930725879856,204623068332997,1688980598900228,14012122025369431,116784468316023069,977437078888272796,8212186058546599006

%N Number of partitions of n^2+n into parts not greater than n.

%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A206227/b206227.txt">Table of n, a(n) for n = 0..382</a> (first 150 terms from Alois P. Heinz)

%F a(n) = [x^(n^2+n)] Product_{k=1..n} 1/(1 - x^k).

%F a(n) ~ c * d^n / n^2, where d = 9.1533701924541224619485302924013545... = A258268, c = 0.3572966225745094270279188015952797... . - _Vaclav Kotesovec_, Sep 07 2014

%p T:= proc(n, k) option remember;

%p       `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))

%p     end:

%p seq(T(n^2+n, n), n=0..20); # _Vaclav Kotesovec_, May 25 2015 after _Alois P. Heinz_

%t Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n+1)}],{n,0,20}] (* _Vaclav Kotesovec_, May 25 2015 *)

%o (PARI) {a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2+n)))),n^2+n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A173519, A206226, A206240, A107379, A258268.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 05 2012