A323314 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 7*x*A(x) )^n * 3^n / 4^(n+1).

1, 27, 2625, 429195, 95328009, 26290301175, 8582072887881, 3220902003386403, 1363088948866736193, 641495666596787938899, 332204944661961666375393, 187727027521862538450725607, 114965661645391124805612197265, 75859037026020765382177030210443, 53662537374831689572836358288777665, 40519124222573071898287923651933134187, 32530810789422606721939134905409891249177, 27674478227000422349878455201664033007066919

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

G.f. A(x) satisfies the following identities.

(1) 1 = Sum_{n>=0} ( (1+x)^n - 7*x*A(x) )^n * 3^n / 4^(n+1).

(2) 1 = Sum_{n>=0} (1+x)^(n^2) * 3^n / (4 + 21*x*A(x)*(1+x)^n)^(n+1).

Example

G.f.: A(x) = 1 + 27*x + 2625*x^2 + 429195*x^3 + 95328009*x^4 + 26290301175*x^5 + 8582072887881*x^6 + 3220902003386403*x^7 + 1363088948866736193*x^8 + ...
such that
1 = 1/4 + ((1+x) - 7*x*A(x))*3/4^2 + ((1+x)^2 - 7*x*A(x))^2*3^2/4^3 + ((1+x)^3 - 7*x*A(x))^3*3^3/4^4 + ((1+x)^4 - 7*x*A(x))^4*3^4/4^5 + ...
Also,
1 = 1/(4 + 21*x*A(x)) + (1+x)*3/(4 + 21*x*A(x)*(1+x))^2 + (1+x)^4*3^2/(4 + 21*x*A(x)*(1+x)^2)^3 + (1+x)^9*3^3/(4 + 21*x*A(x)*(1+x)^3)^4 + ...

Prog

(PARI) \p120
{A=vector(1); A[1]=1; for(i=1, 20, A = concat(A, 0);
A[#A] = round( Vec( sum(n=0, 1200, ( (1+x +x*O(x^#A))^n - 7*x*Ser(A) )^n * 3^n/4^(n+1)*1.)/21 ) )[#A+1]); A}

Crossrefs

Cf. A301435, A303288, A323315, A323316, A323317, A323318, A323319, A323320, A323321.

Sequence in context: A060629 A287228 A086206 * A295022 A017427 A050644

Adjacent sequences: A323311 A323312 A323313 * A323315 A323316 A323317

Keyword

nonn

Author

Paul D. Hanna, Jan 10 2019

Status

approved