A366732 - OEIS

%I #6 Oct 29 2023 22:02:25

%S 1,2,4,22,108,574,3224,18592,109728,660938,4041900,25034000,156724204,

%T 990127086,6304425800,40416596578,260658078580,1689976752116,

%U 11008752656960,72016455973262,472912945955364,3116243639293972,20599091568973324,136557058462319178,907668022344460584

%N Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (2 - x^(n-1))^(n+1).

%C a(n) = Sum_{k=0..n} A366730(n,k) * 2^k for n >= 0.

%H Paul D. Hanna, <a href="/A366732/b366732.txt">Table of n, a(n) for n = 0..250</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

%F (1) 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (2 - x^(n-1))^(n+1).

%F (2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( A(x)^n * (1 - 2*x^(n+1))^(n-1) ).

%e G.f.: A(x) = 1 + 2*x + 4*x^2 + 22*x^3 + 108*x^4 + 574*x^5 + 3224*x^6 + 18592*x^7 + 109728*x^8 + 660938*x^9 + 4041900*x^10 + 25034000*x^11 + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff( sum(n=-#A,#A, x^n * Ser(A)^n * (2 - x^(n-1))^(n+1) ), #A-2));A[n+1]}

%o for(n=0,40,print1(a(n),", "))

%Y Cf. A366730, A366731, A366733, A366734, A366735.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 29 2023