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A206240 Number of partitions of n^2-n into parts not greater than n.

%I A206240 #25 May 07 2018 18:33:29
%S A206240 1,1,2,7,34,192,1206,8033,55974,403016,2977866,22464381,172388026,
%T A206240 1341929845,10573800028,84192383755,676491536028,5479185281572,
%U A206240 44692412971566,366844007355202,3028143252035976,25123376972033392,209401287806758273,1752674793617241002
%N A206240 Number of partitions of n^2-n into parts not greater than n.
%C A206240 Also the number of partitions of n^2 into exactly n parts. - _Seiichi Manyama_, May 07 2018
%H A206240 Alois P. Heinz and Vaclav Kotesovec, <a href="/A206240/b206240.txt">Table of n, a(n) for n = 0..382</a> (first 150 terms from Alois P. Heinz)
%F A206240 a(n) = [x^(n^2-n)] Product_{k=1..n} 1/(1 - x^k).
%F A206240 a(n) ~ c * d^n / n^2, where d = 9.153370192454122461948530292401354540073... = A258268, c = 0.07005383646855329845970382163053268... . - _Vaclav Kotesovec_, Sep 07 2014
%e A206240 From _Seiichi Manyama_, May 07 2018: (Start)
%e A206240 n | Partitions of n^2 into exactly n parts
%e A206240 --+-------------------------------------------------------
%e A206240 1 | 1.
%e A206240 2 | 3+1 = 2+2.
%e A206240 3 | 7+1+1 = 6+2+1 = 5+3+1 = 5+2+2 = 4+4+1 = 4+3+2 = 3+3+3. (End)
%p A206240 T:= proc(n, k) option remember;
%p A206240       `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
%p A206240     end:
%p A206240 seq(T(n^2-n, n), n=0..20); # _Vaclav Kotesovec_, May 25 2015 after _Alois P. Heinz_
%t A206240 Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n-1)}],{n,0,20}] (* _Vaclav Kotesovec_, May 25 2015 *)
%o A206240 (PARI) {a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2-n)))),n^2-n)}
%o A206240 for(n=0,30,print1(a(n),", "))
%Y A206240 Cf. A173519, A206226, A206227, A107379, A258268.
%K A206240 nonn
%O A206240 0,3
%A A206240 _Paul D. Hanna_, Feb 05 2012