|
%I #11 Sep 20 2019 04:15:40
%S 1,75,8317,1239823,216456376,41175714454,8251690444250,
%T 1713228373452375,365077361327242168,79376343363731999772,
%U 17538231051073300512165,3926523351382832339690135,888819911396229761050640552,203083664214425241278951079860
%N Number of compositions (ordered partitions) of 1 into exactly 4n+1 powers of 1/(n+1).
%H Alois P. Heinz, <a href="/A294983/b294983.txt">Table of n, a(n) for n = 0..416</a>
%F From _Vaclav Kotesovec_, Sep 20 2019: (Start)
%F Recurrence: 3*(n-1)^2*n^3*(n+1)*(2*n + 1)*(3*n + 1)*(3*n + 2)*(1794*n^8 - 24743*n^7 + 137870*n^6 - 403038*n^5 + 671286*n^4 - 647019*n^3 + 349914*n^2 - 97024*n + 10624)*a(n) = 4*(n-1)^2*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*(249366*n^11 - 3033833*n^10 + 13899420*n^9 - 28885850*n^8 + 21978374*n^7 + 11319871*n^6 - 24589048*n^5 + 5121972*n^4 + 6329552*n^3 - 2255040*n^2 - 432000*n + 176256)*a(n-1) - 64*(n-1)*(2*n - 3)*(2*n - 1)*(4*n - 5)*(4*n - 3)*(4*n - 1)*(4*n + 1)*(28704*n^10 - 307085*n^9 + 1172466*n^8 - 1799547*n^7 + 417441*n^6 + 1564368*n^5 - 1050675*n^4 - 368572*n^3 + 336852*n^2 + 24624*n - 25056)*a(n-2) + 256*(2*n - 5)*(2*n - 3)*(2*n - 1)*(4*n - 9)*(4*n - 7)*(4*n - 5)*(4*n - 3)*(4*n - 1)*(4*n + 1)*(1794*n^8 - 10391*n^7 + 14901*n^6 + 5043*n^5 - 16279*n^4 - 396*n^3 + 4872*n^2 + 72*n - 336)*a(n-3).
%F a(n) ~ 2^(8*n + 3/2) / (Pi^(3/2) * n^(3/2)). (End)
%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-> (k-> `if`(n=0, 1, b(k*n+1, 1, 0, n+1)))(4):
%p seq(a(n), n=0..20);
%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 + 1, 1, 0, n + 1]]&[4];
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 21 2018, translated from Maple *)
%Y Row n=4 of A294746.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Nov 12 2017
|
