A326563 G.f. A(x) satisfies: Sum_{n>=0} A(x)^((n-1)^2) * x^n = Sum_{n>=0} (A(x)^(n-2) + 1)^n * x^n.

1, 1, 2, 5, 19, 87, 458, 2650, 16459, 108313, 748530, 5400001, 40494822, 314699916, 2528714512, 20972984213, 179305049748, 1578420640715, 14293912578588, 133053118062054, 1272096858349455, 12483241986776063, 125644852173512082, 1296202208475495630, 13696609133818915790, 148138542880240522733, 1638862202896363931279, 18532964342704899428439, 214086944190169455544290, 2524648814791758862995327

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..250

Formula

G.f. A(x) allows the following sums to be equal:

(1) B(x) = Sum_{n>=0} A(x)^((n-1)^2) * x^n.

(2) B(x) = Sum_{n>=0} (A(x)^(n-2) + 1)^n * x^n.

(3) B(x) = Sum_{n>=0} A(x)^(n*(n-2)) * x^n / (1 - x*A(x)^n)^(n+1).

Example

G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 19*x^4 + 87*x^5 + 458*x^6 + 2650*x^7 + 16459*x^8 + 108313*x^9 + 748530*x^10 + 5400001*x^11 + 40494822*x^12 + ...
such that the following sums are 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/A(x))*x + 2^2*x^2 + (1 + A(x))^3*x^3 + (1 + A(x)^2)^4*x^4 + (1 + A(x)^3)^5*x^5 + (1 + A(x)^4)^6*x^6 + ... + (1 + A(x)^(n-2))^n*x^n + ...
also
B(x) = 1/(1 - x) + 1/A(x)*x/(1 - x*A(x))^2 + x^2/(1 - x*A(x)^2)^3 + A(x)^3*x^3/(1 - x*A(x)^3)^4 + ... + A(x)^(n*(n-2))*x^n/(1 - x*A(x)^n)^(n+1) + ...
where
B(x) = 1 + 2*x + 3*x^2 + 7*x^3 + 26*x^4 + 116*x^5 + 596*x^6 + 3373*x^7 + 20541*x^8 + 132803*x^9 + 903151*x^10 + 6420523*x^11 + 47502514*x^12 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, (Ser(A)^(m-2) + 1)^m*x^m - Ser(A)^((m-1)^2)*x^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A326560, A326561, A326562, A326275, A326287.

Sequence in context: A002851 A354621 A324618 * A316700 A124348 A323389

Adjacent sequences: A326560 A326561 A326562 * A326564 A326565 A326566

Keyword

nonn

Author

Paul D. Hanna, Jul 24 2019

Status

approved