A323315 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 9*x*A(x) )^n * 4^n / 5^(n+1).

1, 44, 7096, 1926724, 711117536, 325957899584, 176862173366416, 110333447177205584, 77614355506352291216, 60715204091160869904064, 52262738608604757586146176, 49089829530793158665498530304, 49969859760169581295921965453056, 54804935053668330661788935789639424, 64439695005477056297527256416094395136, 80874250846078911532650120181881418467904

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 - 9*x*A(x) )^n * 4^n / 5^(n+1).

(2) 1 = Sum_{n>=0} (1+x)^(n^2) * 4^n / (5 + 36*x*A(x)*(1+x)^n)^(n+1).

Example

G.f.: A(x) = 1 + 44*x + 7096*x^2 + 1926724*x^3 + 711117536*x^4 + 325957899584*x^5 + 176862173366416*x^6 + 110333447177205584*x^7 + ...
such that
1 = 1/5 + ((1+x) - 9*x*A(x))*4/5^2 + ((1+x)^2 - 9*x*A(x))^2*4^2/5^3 + ((1+x)^3 - 9*x*A(x))^3*4^3/5^4 + ((1+x)^4 - 9*x*A(x))^4*4^4/5^5 + ...
Also,
1 = 1/(5 + 36*x*A(x)) + (1+x)*4/(5 + 36*x*A(x)*(1+x))^2 + (1+x)^4*4^2/(5 + 36*x*A(x)*(1+x)^2)^3 + (1+x)^9*4^3/(5 + 36*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, 1000, ( (1+x +x*O(x^#A))^n - 9*x*Ser(A) )^n * 4^n/5^(n+1)*1.)/36 ) )[#A+1]); A}

Crossrefs

Cf. A301435, A303288, A323314, A323316, A323317, A323318, A323319, A323320, A323321.

Sequence in context: A268548 A203974 A266853 * A007105 A180842 A172753

Adjacent sequences: A323312 A323313 A323314 * A323316 A323317 A323318

Keyword

nonn

Author

Paul D. Hanna, Jan 10 2019

Status

approved