A300044 G.f. A(x) satisfies: 1 + x = Sum_{n>=0} x^n * (1 - x*A(x)^n)^n.

1, -1, 3, -9, 29, -94, 304, -974, 3095, -9809, 31295, -101459, 335712, -1128103, 3796221, -12567881, 40382747, -126022996, 392875458, -1300518891, 4821378972, -19442392758, 76612026077, -258487216389, 602418545348, -104469005952, -6657105734736, 22616539538599, 115937698147766, -1859472149450154, 11984970173891003, -40825530158762433, -37716293058857145, 1564659680870999322

Offset

0,3

Links

Table of n, a(n) for n=0..33.

Formula

G.f. A(x) satisfies:

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

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

Example

G.f.: A(x) = 1 - x + 3*x^2 - 9*x^3 + 29*x^4 - 94*x^5 + 304*x^6 - 974*x^7 + 3095*x^8 - 9809*x^9 + 31295*x^10 - 101459*x^11 + 335712*x^12 + ...
such that
1 + x = 1 + x*(1-x*A(x)) + x^2*(1-x*A(x)^2)^2 + x^3*(1-x*A(x)^3)^3 + x^4*(1-x*A(x)^4)^4 + x^5*(1-x*A(x)^5)^5 + x^6*(1-x*A(x)^6)^6 + ...
The g.f. also satisfies the series identity:
1 + x = 1/(1-x) - x^2*A(x)/(1-x*A(x))^2 + x^4*A(x)^4/(1-x*A(x)^2)^3 - x^6*A(x)^9/(1-x*A(x)^3)^4 + x^8*A(x)^16/(1-x*A(x)^4)^5 - x^10*A(x)^25/(1-x*A(x)^5)^6 + ...

Prog

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

Crossrefs

Sequence in context: A024744 A024964 A061534 * A356937 A071728 A036550

Adjacent sequences: A300041 A300042 A300043 * A300045 A300046 A300047

Keyword

sign

Author

Paul D. Hanna, Feb 24 2018

Status

approved