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%I #15 Jan 23 2021 23:38:44
%S 1,2,4,9,24,79,331,1803,12954,123983,1592513,27604172,648528166,
%T 20722205191,903019659239,53792176322629,4388683843024734,
%U 491232972054490915,75545748143323475653,15984344095578889888206
%N a(n) = Sum_{k=0..n} binomial(2^k + n-k-1, n-k); equals the row sums of triangle A137153.
%C Matrix inverse of A137153 is A137156.
%F G.f.: Sum_{n>=0} x^n/(1-x)^(2^n). - _Paul D. Hanna_, Sep 15 2009
%F G.f.: Sum_{n>=0} ( (-log(1 - x))^n / n! ) / (1 - 2^n*x). - _Paul D. Hanna_, Jan 23 2021
%t Table[Sum[Binomial[2^(n-k) + k - 1, k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jan 23 2021 *)
%o (PARI) a(n)=sum(k=0,n,binomial(2^k+n-k-1,n-k))
%o (PARI) {a(n)=local(A=sum(k=0,n,x^k/(1-x+x*O(x^n))^(2^k)));polcoeff(A,n)} \\ _Paul D. Hanna_, Sep 15 2009
%Y Cf. A137153, A137155, A137156.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 24 2008