A340943 G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(4*n)/(1 - x*A(x)^n) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(4*n+3)).

1, 1, 5, 45, 482, 5665, 70725, 921174, 12379878, 170435921, 2391736448, 34089385297, 492181254691, 7183748957321, 105830560089572, 1571662656809121, 23504719106546214, 353701665355036178, 5351873694519004045, 81378581395212130011

Offset

0,3

Comments

Equals row k = 4 of rectangular table A340940.

Links

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

Formula

Given g.f. A(x), the following sums are all equal:

(1) B(x) = Sum_{n>=0} x^n*A(x)^(4*n)/(1 - x*A(x)^n),

(2) B(x) = Sum_{n>=0} x^n*A(x)^(3*n)/(1 - x*A(x)^(4*n+1)),

(3) B(x) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(4*n+3)),

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

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

(6) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(4*n^2+4*n) * (1 - x^2*A(x)^(8*n+4)) / ((1 - x*A(x)^(4*n+1))*(1 - x*A(x)^(4*n+3)));

see the example section for the value of B(x).

Example

G.f.: A(x) = 1 + x + 5*x^2 + 45*x^3 + 482*x^4 + 5665*x^5 + 70725*x^6 + 921174*x^7 + 12379878*x^8 + 170435921*x^9 + 2391736448*x^10 + ...
such that
B(x) = 1/(1-x) + x*A(x)^4/(1 - x*A(x)) + x^2*A(x)^8/(1 - x*A(x)^2) + x^3*A(x)^12/(1 - x*A(x)^3) + x^4*A(x)^16/(1 - x*A(x)^4) + ...
and
B(x) = 1/(1 - x*A(x)) + x*A(x)^3/(1 - x*A(x)^5) + x^2*A(x)^6/(1 - x*A(x)^9) + x^3*A(x)^9/(1 - x*A(x)^13) + x^4*A(x)^12/(1 - x*A(x)^17) + ...
also
B(x) = 1/(1 - x*A(x)^3) + x*A(x)/(1 - x*A(x)^7) + x^2*A(x)^2/(1 - x*A(x)^11) + x^3*A(x)^3/(1 - x*A(x)^15) + x^4*A(x)^4/(1 - x*A(x)^19)  + ...
further,
B(x) = 1/(1 - x*A(x)^4) + x/(1 - x*A(x)^5) + x^2/(1 - x*A(x)^6) + x^3/(1 - x*A(x)^7) + x^4/(1 - x*A(x)^8) + ...
where
B(x) = 1 + 2*x + 7*x^2 + 43*x^3 + 380*x^4 + 4032*x^5 + 47234*x^6 + 588683*x^7 + 7657593*x^8 + 102796547*x^9 + 1413743374*x^10 + ...

Prog

(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); H=A; A=concat(A, 0);
H[#A-1] = -polcoeff( sum(m=0, #A, x^m/(1 - x*Ser(A)^(m+4)) ) - sum(m=0, #A, x^m*Ser(A)^m/(1 - x*Ser(A)^(4*m+3)) ), #A)/3; A=H); A[n+1] }
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A340940, A340941, A340942, A340894, A340895, A341376.

Sequence in context: A371756 A209442 A371519 * A199753 A220877 A130976

Adjacent sequences: A340940 A340941 A340942 * A340944 A340945 A340946

Keyword

nonn

Author

Paul D. Hanna, Feb 04 2021

Status

approved