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%I #17 Oct 10 2020 05:14:32
%S 1,2,15,201,3807,93103,2788528,98816388,4043274742,187583369889,
%T 9729671519992,557914167187926,35044465503390938,2392988036211331477,
%U 176493963957191423895,13982630491776175877953,1184241622895183679920962,106774511855374079570593467,10211007157153638802035266227,1032332791948276849592811619207
%N G.f.: Sum_{n>=0} ( (1+x)^n - 1/(1+x)^n )^n.
%C Compare to A319947, the dual to this sequence.
%C G.f. A(x) = (1+x) * B( x/(1+x) ), where B(x) is the g.f. of A319947.
%C a(n) - A319947(n) = 0 (mod 2) for n >= 0.
%H Paul D. Hanna, <a href="/A319466/b319466.txt">Table of n, a(n) for n = 0..300</a>
%F G.f.: Sum_{n>=0} (1+x)^(n^2) * Sum_{k=0..n} (-1)^k * binomial(n,k) / (1+x)^(2*n*k).
%F G.f.: Sum_{n>=0} 1/(1+x)^(n^2) * Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * (1+x)^(2*n*k).
%F a(n) ~ c * d^n * n! / sqrt(n), where d = 5.4666049332127684665699843922982444983683628264382802770893... and c = 0.3563391278539240852770166562386253680399190992740998... - _Vaclav Kotesovec_, Oct 10 2020
%e G.f.: A(x) = 1 + 2*x + 15*x^2 + 201*x^3 + 3807*x^4 + 93103*x^5 + 2788528*x^6 + 98816388*x^7 + 4043274742*x^8 + 187583369889*x^9 + ...
%e such that
%e A(x) = 1 + ((1+x) - 1/(1+x)) + ((1+x)^2 - 1/(1+x)^2)^2 + ((1+x)^3 - 1/(1+x)^3)^3 + ((1+x)^4 - 1/(1+x)^4)^4 + ((1+x)^5 - 1/(1+x)^5)^5 + ...
%e Equivalently,
%e A(x) = 1 +
%e ((1+x) - 1/(1+x)) +
%e ((1+x)^4 - 2 + 1/(1+x)^4) +
%e ((1+x)^9 - 3*(1+x)^3 + 3/(1+x)^3 - 1/(1+x)^9) +
%e ((1+x)^16 - 4*(1+x)^8 + 6 - 4/(1+x)^8 + 1/(1+x)^16) +
%e ((1+x)^25 - 5*(1+x)^15 + 10*(1+x)^5 - 10/(1+x)^5 + 5/(1+x)^15 - 1/(1+x)^25) +
%e ((1+x)^36 - 6*(1+x)^24 + 15*(1+x)^12 - 20 + 15/(1+x)^12 - 6/(1+x)^24 + 1/(1+x)^36) + ...
%o (PARI) {a(n) = my(A=1, X=x + x*O(x^n)); A = sum(m=0,n, ((1+x)^m - 1/(1+X)^m)^m );polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A319947.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 28 2018
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