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%I #17 Sep 20 2019 05:26:26
%S 1,1,10,4245,216456376,2713420774885145,14138484434475011392912026,
%T 46050764886573707269872023694736134925,
%U 131223281654667714701311635640432890136981994039662720,435699237793484726791774188056400878106883117166142375354233228879800569
%N Number of compositions (ordered partitions) of 1 into exactly n^2+1 powers of 1/(n+1).
%H Alois P. Heinz, <a href="/A294747/b294747.txt">Table of n, a(n) for n = 0..26</a>
%F a(n) = [x^((n+1)^n)] (Sum_{j=0..n^2+1} x^((n+1)^j))^(n^2+1) for n>0, a(0) = 1.
%F a(n) = A294746(n,n).
%F a(n) ~ exp(-1/12) * n^(n^2 - n/2 + 2) / (2*Pi)^((n-1)/2). - _Vaclav Kotesovec_, Sep 20 2019
%e a(0) = 1: [1].
%e a(1) = 1: [1/2,1/2].
%e a(2) = 10 = binomial(5,2): [1/3,1/3,1/9,1/9,1/9], [1/3,1/9,1/3,1/9,1/9], [1/3,1/9,1/9,1/3,1/9], [1/3,1/9,1/9,1/9,1/3], [1/9,1/3,1/3,1/9,1/9], [1/9,1/3,1/9,1/3,1/9], [1/9,1/3,1/9,1/9,1/3], [1/9,1/9,1/3,1/3,1/9], [1/9,1/9,1/3,1/9,1/3], [1/9,1/9,1/9,1/3,1/3].
%p b:= proc(n, r, p, k) option remember;
%p `if`(n<r, 0, `if`(r=0, `if`(n=0, p!, 0), add(
%p b(n-j, k*(r-j), p+j, k)/j!, j=0..min(n, r))))
%p end:
%p a:= n-> `if`(n=0, 1, b(n^2+1, 1, 0, n+1)):
%p seq(a(n), n=0..10);
%t b[n_, r_, p_, k_] := b[n, r, p, k] = If[n < r, 0, If[r == 0, If[n == 0, p!, 0], Sum[b[n - j, k*(r - j), p + j, k]/j!, {j, 0, Min[n, r]}]]];
%t a[n_] := If[n == 0, 1, b[n^2 + 1, 1, 0, n + 1]];
%t Table[a[n], {n, 0, 10}] (* _Jean-François Alcover_, May 21 2018, translated from Maple *)
%Y Main diagonal of A294746.
%Y Cf. A002522.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Nov 07 2017
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