A323319 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 17*x*A(x) )^n * 8^n / 9^(n+1).

1, 152, 87760, 85439240, 113151839104, 186152435786240, 362548564958149696, 811847325733606058048, 2049967057729258844550208, 5756221555712461523954507264, 17785396518936498493080842349568, 59963943179536216027803213130483712, 219093913413498532617018883655015864320, 862506026576114820987041351988191302565888, 3640101913203153345185251232178995247004487680, 16397805545827151302219567488776238270687543337472

Offset

0,2

Links

Table of n, a(n) for n=0..15.

Formula

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

(1) 1 = Sum_{n>=0} ( (1+x)^n - 17*x*A(x) )^n * 8^n / 9^(n+1).

(2) 1 = Sum_{n>=0} (1+x)^(n^2) * 8^n / (9 + 136*x*A(x)*(1+x)^n)^(n+1).

Example

G.f.: A(x) = 1 + 152*x + 87760*x^2 + 85439240*x^3 + 113151839104*x^4 + 186152435786240*x^5 + 362548564958149696*x^6 + 811847325733606058048*x^7 + ...
such that
1 = 1/9 + ((1+x) - 17*x*A(x))*8/9^2 + ((1+x)^2 - 17*x*A(x))^2*8^2/9^3 + ((1+x)^3 - 17*x*A(x))^3*8^3/9^4 + ((1+x)^4 - 17*x*A(x))^4*8^4/9^5 + ...
Also,
1 = 1/(9 + 136*x*A(x)) + (1+x)*8/(9 + 136*x*A(x)*(1+x))^2 + (1+x)^4*8^2/(9 + 136*x*A(x)*(1+x)^2)^3 + (1+x)^9*8^3/(9 + 136*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, 2200, ( (1+x +x*O(x^#A))^n - 17*x*Ser(A) )^n * 8^n/9^(n+1)*1.)/136 ) )[#A+1]); A}

Crossrefs

Cf. A301435, A303288, A323314, A323315, A323316, A323317, A323318, A323320, A323321.

Sequence in context: A265861 A208484 A335091 * A157881 A370312 A099117

Adjacent sequences: A323316 A323317 A323318 * A323320 A323321 A323322

Keyword

nonn

Author

Paul D. Hanna, Jan 10 2019

Status

approved