A204062 Expansion of g.f.: exp( Sum_{n>=1} A002203(n)^2 * x^n/n ) where A002203 are the companion Pell numbers.

1, 4, 26, 148, 867, 5048, 29428, 171512, 999653, 5826396, 33958734, 197925996, 1153597255, 6723657520, 39188347880, 228406429744, 1331250230601, 7759094953844, 45223319492482, 263580822001028, 1536261612513707, 8953988853081192, 52187671505973468

Offset: 0

Author

Paul D. Hanna, Jan 10 2012

Keywords

• nonn
• easy

Examples

			G.f.: A(x) = 1 + 4*x + 26*x^2 + 148*x^3 + 867*x^4 + 5048*x^5 + ...
where
log(A(x)) = 2^2*x + 6^2*x^2/2 + 14^2*x^3/3 + 34^2*x^4/4 + 82^2*x^5/5 + 198^2*x^6/6 + 478^2*x^7/7 + ... + A002203(n)^2*x^n/n + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (4,10,4,-1).

Crossrefs

Cf. A000129, A002203, A026933, A204061, A212442.

Programs

Magma

I:=[1,4,26,148]; [n le 4 select I[n] else 4*Self(n-1) +10*Self(n-2) +4*Self(n-3) -Self(n-4): n in [1..31]]; // G. C. Greubel, May 25 2021

Mathematica

LinearRecurrence[{4,10,4,-1},{1,4,26,148},30] (* Vincenzo Librandi, Feb 12 2012 *)
Table[(Fibonacci[2*n+4, 2] + 2*(-1)^n*(n+2))/16, {n, 0, 30}] (* G. C. Greubel, May 25 2021 *)

PARI

{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^2*x^k/k)+x*O(x^n)), n)}

Sage

[(lucas_number1(2*n+4,2,-1) +2*(-1)^n*(n+2))/16 for n in (0..30)] # G. C. Greubel, May 25 2021

Formula

G.f.: 1/((1+x)^2*(1-6*x+x^2)).

Self-convolution of A026933.

Self-convolution 4th power of A204061.

a(n) = Pell(n-1)^2 + a(n-2) where Pell(n) = A000129(n).

a(n) = (1/8)*(A001109(n+2) + (-1)^n*(n+2)). - Bruno Berselli, Jan 10 2012

a(n) = (1/16)*(A000129(2*n+4) + 2*(-1)^n*(n+2)). - G. C. Greubel, May 25 2021