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A194583 Triangle T(n,k) with T(n,0)=1 and T(n,k) = (2^(n+1)-2^k)*T(n,k-1) + T(n+1,k-1) otherwise.

%I A194583 #17 Dec 23 2024 14:53:42
%S A194583 1,1,3,1,7,43,1,15,211,2619,1,31,931,26251,654811,1,63,3907,234795,
%T A194583 13255291,662827803,1,127,16003,1985131,238658491,26961325147,
%U A194583 2699483026843,1,255,64771,16323819,4050110011,973958217435,220115609012251,44102911693372059,1,511,260611,132393451,66733574971,33115631264731,15928113739803931,7200501591899676571,2886238576935227688091
%N A194583 Triangle T(n,k) with T(n,0)=1 and T(n,k) = (2^(n+1)-2^k)*T(n,k-1) + T(n+1,k-1) otherwise.
%H A194583 G. Helms, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2011-August/015257.html">Number array not found in OEIS</a>, SeqFan list Aug 27 2011
%F A194583 T(n,1) = A000225(n+1).
%F A194583 T(n,2) = (2^(n+1)-4)*(2^(n+1)-1) + 2^(n+2) - 1.
%F A194583 T(n,k) = -Sum_{j=1..k+1} A158474(k+1,j)*T(n-j,k) assuming the symmetric extension T(n,k) = T(k,n).
%e A194583 The triangle starts in row n=0 as
%e A194583   1;
%e A194583   1,  3;
%e A194583   1,  7,  43;
%e A194583   1, 15, 211,  2619;
%e A194583   1, 31, 931, 26251, 654811;
%p A194583 A194583 := proc(n,k) option remember; if n=0 or k=0 then 1; elif k> n then
%p A194583 return procname(k,n); else (2^(n+1)-2^k)*procname(n,k-1)+procname(n+1,k-1) ; end if;
%p A194583 end proc:
%t A194583 t[_, 0] = 1; t[n_, k_] := t[n, k] = (2^(n+1)-2^k)*t[n, k-1]+t[n+1, k-1]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 10 2014 *)
%K A194583 nonn,tabl
%O A194583 0,3
%A A194583 _R. J. Mathar_, Aug 29 2011