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A153280 Eigensequence of triangle A153279.

%I A153280 #22 Jun 02 2025 01:16:32
%S A153280 1,3,15,165,4785,397155,97302975,71128474725,155700231173025,
%T A153280 1021860617188563075,20115326249356864131375,
%U A153280 1187830130350772183821825125,210422919761508941591852499068625,111827787746815596446398867662527275875
%N A153280 Eigensequence of triangle A153279.
%H A153280 Harvey P. Dale, <a href="/A153280/b153280.txt">Table of n, a(n) for n = 0..65</a>
%F A153280 Given triangle A153279, let a new triangle = M shifted down one row, inserting a "1" in (0,0). Triangle equals lim_{n->oo} M^n.
%F A153280 a(n+1) = a(n)*(2+3^n), a(0) = 1. - _Philippe Deléham_, Sep 27 2014
%F A153280 a(n) ~ c * 3^(n*(n-1)/2), where c = QPochhammer(-2, 1/3) = 6.80914656805984199... - _Vaclav Kotesovec_, Jan 22 2023
%e A153280 Triangle M =
%e A153280 1;
%e A153280 1;
%e A153280 2, 1;
%e A153280 4, 2, 3;
%e A153280 8, 4, 6, 9;
%e A153280 16, 8, 12, 18, 27;
%e A153280 ...
%e A153280 M^n rapidly converges to this sequence with sufficiently large n.
%e A153280 a(0) = 1, a(1) = 1*(2+3^0) = 3, a(2) = 3*(2+3^1) = 15, a(3) = 15*(2+3^2) = 165, a(4) = 165*(2+3^3) = 4785, ... - _Philippe Deléham_, Sep 27 2014
%t A153280 RecurrenceTable[{a[n+1] == a[n]*(2 + 3^n), a[0] == 1}, a, {n, 0, 15}] (* _Vaclav Kotesovec_, Jan 22 2023 *)
%t A153280 Table[2^n * QPochhammer[-1/2, 3, n], {n, 0, 15}] (* _Vaclav Kotesovec_, Jan 22 2023 *)
%t A153280 nxt[{n_,a_}]:={n+1,a(2+3^n)}; NestList[nxt,{0,1},20][[;;,2]] (* _Harvey P. Dale_, Mar 28 2024 *)
%Y A153280 Cf. A153279, A057735.
%K A153280 nonn
%O A153280 0,2
%A A153280 _Gary W. Adamson_, Dec 23 2008
%E A153280 More terms from _Philippe Deléham_, Sep 27 2014