A323316 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 11*x*A(x) )^n * 5^n / 6^(n+1).

1, 65, 15685, 6376505, 3524871325, 2420187902975, 1967093055766825, 1838251199473028225, 1937082794808580188025, 2269921874941072916242625, 2926922052137279952439869625, 4118264067683762888405147993375, 6279611163775388892921689107812625, 10316794138820163374949788420225125625, 18170957626950430345183391610737313950125, 34161178486729901360568404660435153779920125

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 - 11*x*A(x) )^n * 5^n / 6^(n+1).

(2) 1 = Sum_{n>=0} (1+x)^(n^2) * 5^n / (6 + 55*x*A(x)*(1+x)^n)^(n+1).

Example

G.f.: A(x) = 1 + 65*x + 15685*x^2 + 6376505*x^3 + 3524871325*x^4 + 2420187902975*x^5 + 1967093055766825*x^6 + 1838251199473028225*x^7 + ...
such that
1 = 1/6 + ((1+x) - 11*x*A(x))*5/6^2 + ((1+x)^2 - 11*x*A(x))^2*5^2/6^3 + ((1+x)^3 - 11*x*A(x))^3*5^3/6^4 + ((1+x)^4 - 11*x*A(x))^4*5^4/6^5 + ...
Also,
1 = 1/(6 + 55*x*A(x)) + (1+x)*5/(6 + 55*x*A(x)*(1+x))^2 + (1+x)^4*5^2/(6 + 55*x*A(x)*(1+x)^2)^3 + (1+x)^9*5^3/(6 + 55*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 - 11*x*Ser(A) )^n * 5^n/6^(n+1)*1.)/55 ) )[#A+1]); A}

Crossrefs

Cf. A301435, A303288, A323314, A323315, A323317, A323318, A323319, A323320, A323321.

Sequence in context: A176370 A093265 A264541 * A120801 A308697 A368890

Adjacent sequences: A323313 A323314 A323315 * A323317 A323318 A323319

Keyword

nonn

Author

Paul D. Hanna, Jan 10 2019

Status

approved