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A090366 Shifts 1 place left under the INVERT transform of the BINOMIAL transform of the self-convolution of this sequence.

%I A090366 #15 Aug 25 2024 18:45:10
%S A090366 1,1,4,21,131,917,6988,56965,491240,4447558,42048457,413473928,
%T A090366 4215959294,44469487070,484303175837,5437300482651,62848069403649,
%U A090366 747063566345320,9123406697372938,114370704441951620,1470590692488141315,19381056189738194070
%N A090366 Shifts 1 place left under the INVERT transform of the BINOMIAL transform of the self-convolution of this sequence.
%H A090366 Alois P. Heinz, <a href="/A090366/b090366.txt">Table of n, a(n) for n = 0..200</a>
%F A090366 G.f.: A(x) = 1/(1 - A(x/(1-x))^2*x/(1-x) ).
%p A090366 bintr:= proc(p) local b; b:= proc(n) option remember;
%p A090366            add(p(k) *binomial(n,k), k=0..n) end
%p A090366         end:
%p A090366 invtr:= proc(p) local b; b:= proc(n) option remember;
%p A090366            `if`(n<1, 1, add(b(n-i) *p(i-1), i=1..n+1)) end
%p A090366         end:
%p A090366 b:= invtr(bintr(n-> add(a(i)*a(n-i), i=0..n))):
%p A090366 a:= n-> `if`(n<0, 0, b(n-1)):
%p A090366 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jun 28 2012
%t A090366 m = 30; A[_] = 1; Do[A[x_] = 1/(1 - A[x/(1-x)]^2*(x/(1-x))) + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] (* _Jean-François Alcover_, Jun 04 2018 *)
%o A090366 (PARI) {a(n)=local(A); if(n<0,0,A=1+x+x*O(x^n); for(k=1,n,B=subst(A^2,x,x/(1-x))/(1-x)+x*O(x^n); A=1+x*A*B);polcoeff(A,n,x))}
%Y A090366 Cf. A090365, A090367.
%K A090366 nonn
%O A090366 0,3
%A A090366 _Paul D. Hanna_, Nov 26 2003