A326558 G.f. A(x) satisfies: Sum_{n>=0} A(x)^((n-1)^2) * x^n = Sum_{n>=0} ((1+x)^(n-1) + 1)^n * x^n.
1, 1, 3, 10, 33, 126, 535, 2493, 12569, 68092, 394029, 2423257, 15767827, 108113333, 778198815, 5859990774, 46016573248, 375739918599, 3181928792057, 27881971943496, 252291917965337, 2353166240022595, 22588471117451579, 222844380918633385, 2256632985093301978, 23430724178641674376, 249198423317854694260, 2712372210346517891943
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 33*x^4 + 126*x^5 + 535*x^6 + 2493*x^7 + 12569*x^8 + 68092*x^9 + 394029*x^10 + 2423257*x^11 + 15767827*x^12 + ... such that the following series are all equal B(x) = A(x) + x + A(x)*x^2 + A(x)^4*x^3 + A(x)^9*x^4 + A(x)^16*x^5 + A(x)^25*x^6 + A(x)^36*x^7 + A(x)^49*x^8 + ... + A(x)^((n-1)^2) * x^n + ... and B(x) = 1 + (1 + 1)*x + (1 + (1+x))^2*x^2 + (1 + (1+x)^2)^3*x^3 + (1 + (1+x)^3)^4*x^4 + (1 + (1+x)^4)^5*x^5 + ... + (1 + (1+x)^(n-1))^n*x^n + ... also B(x) = 1/(1 - x) + x/(1 - x*(1+x))^2 + (1+x)^2*x^2/(1 - x*(1+x)^2)^3 + (1+x)^6*x^3/(1 - x*(1+x)^3)^4 + ... + (1+x)^(n*(n-1))*x^n/(1 - x*(1+x)^n)^(n+1) + ... where B(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 41*x^4 + 164*x^5 + 728*x^6 + 3546*x^7 + 18679*x^8 + 105445*x^9 + 633198*x^10 + 4021124*x^11 + 26876020*x^12 + ...
Crossrefs
Cf. A326557.
Programs
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PARI
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = polcoeff( sum(n=0,#A, ((1+x)^(n-1) + 1 +x*O(x^#A))^n *x^n - Ser(A)^((n-1)^2) *x^n ),#A-1));A[n+1]} for(n=0,30,print1(a(n),", "))
Formula
G.f. A(x) satisfies:
(1) Sum_{n>=0} A(x)^((n-1)^2) * x^n = Sum_{n>=0} ((1+x)^(n-1) + 1)^n * x^n.
(2) Sum_{n>=0} A(x)^((n-1)^2) * x^n = Sum_{n>=0} (1+x)^(n*(n-1)) * x^n / (1 - x*(1+x)^n)^(n+1).