A351772 - OEIS

A351772

G.f. A(x) = Sum_{n>=0} x^n*F(x)^n/(1 - x*F(x)^(n+4)), where F(x) is the g.f. of A350117.

1, 2, 8, 51, 442, 4534, 51182, 613806, 7675397, 98971497, 1306630823, 17575262387, 240012293969, 3319086310532, 46386983964844, 654176372802786, 9297814382343636, 133052398800475776, 1915431497096942109, 27721644693710659258

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OFFSET

0,2

LINKS

Table of n, a(n) for n=0..19.

FORMULA

The g.f. A(x) of this sequence can be determined from the g.f. F(x) of A350117 as follows.

(1) A(x) = Sum_{n>=0} x^n*F(x)^(1*n) / (1 - x*F(x)^(1*n+4));

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

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

(4) A(x) = Sum_{n>=0} x^n*F(x)^(4*n) / (1 - x*F(x)^(1*n+1));

(5) A(x) = Sum_{n>=0} x^(2*n) * F(x)^(n^2+5*n) * (1 - x^2*F(x)^(2*n+5)) / ((1 - x*F(x)^(n+1))*(1 - x*F(x)^(n+4))),

(6) A(x) = Sum_{n>=0} x^(2*n) * F(x)^(3*n^2+5*n) * (1 - x^2*F(x)^(6*n+5)) / ((1 - x*F(x)^(3*n+2))*(1 - x*F(x)^(3*n+3)));

see the example section for the series expansion of F(x).

EXAMPLE

G.f.: A(x) = 1 + 2*x + 8*x^2 + 51*x^3 + 442*x^4 + 4534*x^5 + 51182*x^6 + 613806*x^7 + 7675397*x^8 + 98971497*x^9 + 1306630823*x^10 + ...

such that

(1) A(x) = 1/(1 - x*F(x)^4) + x*F(x)^1/(1 - x*F(x)^5) + x^2*F(x)^2/(1 - x*F(x)^6) + x^3*F(x)^3/(1 - x*F(x)^7) + x^4*F(x)^4/(1 - x*F(x)^8) + ...

(2) A(x) = 1/(1 - x*F(x)^3) + x*F(x)^2/(1 - x*F(x)^6) + x^2*F(x)^4/(1 - x*F(x)^9) + x^3*F(x)^6/(1 - x*F(x)^12) + x^4*F(x)^8/(1 - x*F(x)^15) + ...

(3) A(x) = 1/(1 - x*F(x)^2) + x*F(x)^3/(1 - x*F(x)^5) + x^2*F(x)^6/(1 - x*F(x)^8) + x^3*F(x)^9/(1 - x*F(x)^11) + x^4*F(x)^12/(1 - x*F(x)^14) + ...

(4) A(x) = 1/(1 - x*F(x)^1) + x*F(x)^4/(1 - x*F(x)^2) + x^2*F(x)^8/(1 - x*F(x)^3) + x^3*F(x)^12/(1 - x*F(x)^4) + x^4*F(x)^16/(1 - x*F(x)^5) + ...

where

F(x) = 1 + x + 5*x^2 + 43*x^3 + 443*x^4 + 5009*x^5 + 60104*x^6 + 751778*x^7 + 9696036*x^8 + 128037209*x^9 + 1722632206*x^10 + ... + A350117(n)*x^n + ...

PROG

(PARI) {a(n) = my(F=[1, 1, 0]); for(i=0, n, F=concat(F, 0);

A1 = sum(m=0, #F, x^m*Ser(F)^(2*m)/(1 - x*Ser(F)^(3*m+3)) );

A2 = sum(m=0, #F, x^m*Ser(F)^(4*m)/(1 - x*Ser(F)^(1*m+1)) );

F[#F-1] = polcoeff((A1 - A2)/2, #F); ); polcoeff(A1, n)}

for(n=0, 30, print1(a(n), ", "))

(PARI) {a(n) = my(F=[1, 1, 0]); for(i=0, n, F=concat(F, 0);

A1 = sum(m=0, #F, x^m*Ser(F)^(3*m)/(1 - x*Ser(F)^(3*m+2)) );

A2 = sum(m=0, #F, x^m*Ser(F)^(1*m)/(1 - x*Ser(F)^(1*m+4)) );

F[#F-1] = polcoeff((A1 - A2)/2, #F); ); polcoeff(A1, n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS