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A136557 a(n) = Sum_{k=0..n} binomial(2^k + n-k-1, k).

%I A136557 #10 Mar 15 2021 21:30:27
%S A136557 1,2,6,45,1436,171836,68149425,89431630740,396956313475102,
%T A136557 6099399658235428041,331007760926212498510464,
%U A136557 64484289650612910347505873728,45677712418497545460138258802186905
%N A136557 a(n) = Sum_{k=0..n} binomial(2^k + n-k-1, k).
%H A136557 G. C. Greubel, <a href="/A136557/b136557.txt">Table of n, a(n) for n = 0..50</a>
%F A136557 Equals antidiagonal sums of square array A136555.
%F A136557 G.f.: A(x) = Sum_{n>=0} (1+2^n*x)^-1 * (1-x-2^n*x^2)^-1 * log(1+2^n*x)^n / n!.
%F A136557 a(n) ~ 2^(n^2) / n!. - _Vaclav Kotesovec_, Jul 02 2016
%p A136557 A136557:= n-> add(binomial(2^k +n-k-1, k), k=0..n); seq(A136557(n), n=0..20); # _G. C. Greubel_, Mar 15 2021
%t A136557 Table[Sum[Binomial[2^k+n-k-1, k], {k, 0, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Jul 02 2016 *)
%o A136557 (PARI) a(n)=sum(k=0,n,binomial(2^k+n-k-1,k))
%o A136557 (PARI) /* As coefficient of x^n in the g.f.: */ {a(n)=polcoeff(sum(i=0,n,((1+2^i*x+x*O(x^n))*(1-x-2^i*x^2))^-1*log(1+2^i*x+x*O(x^n))^i/i!),n)}
%o A136557 (Sage) [sum(binomial(2^k +n-k-1, k) for k in (0..n)) for n in (0..20)] # _G. C. Greubel_, Mar 15 2021
%o A136557 (Magma) [(&+[Binomial(2^k +n-k-1, k): k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Mar 15 2021
%Y A136557 Cf. A136555, A136556.
%K A136557 nonn
%O A136557 0,2
%A A136557 _Paul D. Hanna_, Jan 07 2008