A325575 G.f. A(x) satisfies: 1/(1-x) = Sum_{n>=0} x^n * ((1+x)^n - A(x))^n, where A(0) = 0.

1, 2, 5, 15, 59, 262, 1307, 7074, 41012, 252187, 1632799, 11074271, 78360644, 576612899, 4400858604, 34762324434, 283657506679, 2387255705823, 20692499748472, 184498910151279, 1690257693844243, 15894461099811120, 153272602343966985, 1514370059327255381, 15317844239550849137, 158501683635111855424, 1676615643571796233437, 18117887771586127697132, 199886514026342226648647

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:

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

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

Example

G.f.: A(x) = x + 2*x^2 + 5*x^3 + 15*x^4 + 59*x^5 + 262*x^6 + 1307*x^7 + 7074*x^8 + 41012*x^9 + 252187*x^10 + 1632799*x^11 + 11074271*x^12 + ...
such that
1/(1-x) = 1 + x*((1+x) - A(x)) + x^2*((1+x)^2 - A(x))^2 + x^3*((1+x)^3 - A(x))^3 + x^4*((1+x)^4 - A(x))^4 + x^5*((1+x)^5 - A(x))^5 + x^6*((1+x)^6 - A(x))^6 + ...
also
1/(1-x) = 1/(1 + x*A(x)) + x*(1+x)/(1 + x*(1+x)*A(x))^2 + x^2*(1+x)^4/(1 + x*(1+x)^2*A(x))^3  + x^3*(1+x)^9/(1 + x*(1+x)^3*A(x))^4 + x^4*(1+x)^16/(1 + x*(1+x)^4*A(x))^5 + x^5*(1+x)^25/(1 + x*(1+x)^5*A(x))^6 + ...

Prog

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

Crossrefs

Cf. A325577, A307940.

Sequence in context: A078792 A208808 A266682 * A332248 A030934 A030922

Adjacent sequences: A325572 A325573 A325574 * A325576 A325577 A325578

Keyword

nonn

Author

Paul D. Hanna, May 16 2019

Status

approved