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%I #29 Feb 12 2024 08:39:59
%S 1,7,157,5419,220561,9763807,454635973,21894817147,1080094827649,
%T 54250971690007,2763339510402637,142338478909290187,
%U 7399210542653679985,387578046480606144079,20433042381373273363477,1083193405190852829195259,57697563083258107660231681
%N a(n) = g(n,n,n) where g(a, b, c) is defined as follows: if a = 0 or b = 0 or c = 0 then return 1 otherwise return g(a, b, c-1) + g(a, b-1, c) + g(a-1, b, c) + g(a, b-1, c-1) + g(a-1, b, c-1) + g(a-1, b-1, c) + g(a-1, b-1, c-1).
%C A generalization of the recurrence in A001850. The original description of this sequence was the same as that of A126086. The correct explanation for these terms was provided by _Nick Hobson_, Mar 03 2007.
%H Alois P. Heinz, <a href="/A115866/b115866.txt">Table of n, a(n) for n = 0..500</a>
%F D-finite with recurrence: 2*(n-1)^2*(2*n-1)*(243*n^5 - 3159*n^4 + 16254*n^3 - 41325*n^2 + 51838*n - 25620)*a(n) = (53703*n^8 - 887922*n^7 + 6273882*n^6 - 24692601*n^5 + 59070956*n^4 - 87717383*n^3 + 78694087*n^2 - 38816698*n + 8003688)*a(n-1) + (94527*n^8 - 1549611*n^7 + 10848681*n^6 - 42278007*n^5 + 100087538*n^4 - 147021644*n^3 + 130465402*n^2 - 63678226*n + 13003980)*a(n-2) - (31833*n^8 - 541890*n^7 + 3945213*n^6 - 16007835*n^5 + 39486422*n^4 - 60435299*n^3 + 55812796*n^2 - 28273516*n + 5965068)*a(n-3) + (n-3)*(3159*n^7 - 48114*n^6 + 301212*n^5 - 1002003*n^4 + 1908157*n^3 - 2073535*n^2 + 1184960*n - 272792)*a(n-4) - 2*(n-4)^2*(n-3)*(243*n^5 - 1944*n^4 + 6048*n^3 - 9087*n^2 + 6529*n - 1769)*a(n-5). - _Vaclav Kotesovec_, Nov 27 2016
%F a(n) ~ (12*2^(2/3)+15*2^(1/3)+19)^n / (2^(4/3)*3^(1/2)*Pi*n). - _Vaclav Kotesovec_, Nov 27 2016
%p g():= seq(convert(n, base, 2)[1..3], n=9..15):
%p b:= proc(l) option remember;
%p `if`(l[1]=0, 1, add(b(sort(l-h)), h=g()))
%p end:
%p a:= n-> b([n$3]):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Oct 14 2015
%t g[] = Table[Reverse[IntegerDigits[n, 2]][[;; 3]], {n, 2^3 + 1, 2^4 - 1}];
%t b[l_] := b[l] = If[l[[1]] == 0, 1, Sum[b[Sort[l - h]], {h, g[]}]];
%t a[n_] := b[Table[n, {3}]];
%t a /@ Range[0, 25] (* _Jean-François Alcover_, Apr 25 2020, after _Alois P. Heinz_ *)
%Y Cf. A001850, A126086.
%Y Column k=3 of A263159.
%K nonn
%O 0,2
%A _Al Zimmermann_, Apr 02 2006
%E Edited by _N. J. A. Sloane_ following email from _Nick Hobson_, Mar 03 2007
%E More terms from _Alois P. Heinz_, Sep 30 2015
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