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A292463 Number of partitions of n with n kinds of 1.

%I A292463 #29 Mar 17 2024 10:45:48
%S A292463 1,1,4,14,51,188,702,2644,10026,38223,146359,562456,2168134,8379539,
%T A292463 32459199,125984039,489837300,1907490728,7438346255,29042470132,
%U A292463 113522618066,444199913556,1739735079466,6819657196928,26753893533257,105034060120469,412637434996367
%N A292463 Number of partitions of n with n kinds of 1.
%H A292463 Alois P. Heinz, <a href="/A292463/b292463.txt">Table of n, a(n) for n = 0..1664</a>
%F A292463 a(n) = [x^n] 1/(1-x)^n * 1/Product_{j=2..n} (1-x^j).
%F A292463 a(n) is n-th term of the Euler transform of n,1,1,1,... .
%F A292463 a(n) ~ c * 4^n / sqrt(n), where c = QPochhammer[-1, 1/2] / (8*sqrt(Pi) * QPochhammer[1/4, 1/4]) = 0.48841139329043831428669851139824427133317... - _Vaclav Kotesovec_, Sep 19 2017
%F A292463 Equivalently, c = 1/(4*sqrt(Pi)*QPochhammer(1/2)). - _Vaclav Kotesovec_, Mar 17 2024
%e A292463 a(2) = 4: 2, 1a1a, 1a1b, 1b1b.
%p A292463 b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
%p A292463       binomial(k+n-1, n), add(b(n-i*j, i-1, k), j=0..n/i))
%p A292463     end:
%p A292463 a:= n-> b(n$3):
%p A292463 seq(a(n), n=0..30);
%p A292463 # second Maple program:
%p A292463 b:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A292463       (numtheory[sigma](j)+k-1)*b(n-j, k), j=1..n)/n)
%p A292463     end:
%p A292463 a:= n-> b(n$2):
%p A292463 seq(a(n), n=0..30);
%p A292463 # third Maple program:
%p A292463 b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=1,
%p A292463       combinat[numbpart](n), b(n-1, k) +b(n, k-1)))
%p A292463     end:
%p A292463 a:= n-> b(n$2):
%p A292463 seq(a(n), n=0..30);
%t A292463 Table[SeriesCoefficient[1/(1-x)^(n-1) * Product[1/(1-x^k), {k,1,n}], {x,0,n}], {n,0,30}] (* _Vaclav Kotesovec_, Sep 19 2017 *)
%Y A292463 Main diagonal of A292508.
%Y A292463 Cf. A000041, A014329, A292462, A292503, A292507, A292541.
%K A292463 nonn
%O A292463 0,3
%A A292463 _Alois P. Heinz_, Sep 16 2017