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A325575 G.f. A(x) satisfies: 1/(1-x) = Sum_{n>=0} x^n * ((1+x)^n - A(x))^n, where A(0) = 0. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 29 10:06 EDT 2020. Contains 337428 sequences. (Running on oeis4.)