A366731 - OEIS

%I #7 Oct 29 2023 22:02:18

%S 1,1,0,2,2,6,19,41,99,307,750,2062,5776,15674,43700,123729,345728,

%T 982580,2801615,7994268,22953104,66128105,190846074,552959720,

%U 1605817449,4673526011,13635237816,39860703465,116739997283,342538898105,1006709394181,2963267980415,8735388348630

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

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

%H Paul D. Hanna, <a href="/A366731/b366731.txt">Table of n, a(n) for n = 0..300</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 * (1 - x^(n-1))^(n+1).

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

%e G.f.: A(x) = 1 + x + 2*x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 41*x^7 + 99*x^8 + 307*x^9 + 750*x^10 + 2062*x^11 + 5776*x^12 + 15674*x^13 + 43700*x^14 + ...

%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 * (1 - x^(n-1))^(n+1) ), #A-2));A[n+1]}

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

%Y Cf. A366730, A366732, A366733, A366734, A366735.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Oct 29 2023