A394707 Coefficient of x^n in the expansion of ( (1+x)^2 * (1+x^2) )^n.

1, 2, 8, 38, 188, 952, 4904, 25580, 134684, 714296, 3810208, 20420446, 109870820, 593109896, 3210820656, 17424460888, 94761252636, 516321274936, 2817962700128, 15402786124216, 84303705432688, 461978027703776, 2534398515953576, 13917634474247388, 76499074055717828

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(2*n,n-2*k).

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x^2)) ). See A369439.

3*n * (3*n - 2) * (3*n - 1) * (57*n^3 - 300*n^2 + 505*n - 274) * a(n) = 8 * (1254*n^6 - 8481*n^5 + 21988*n^4 - 27874*n^3 + 18107*n^2 - 5672*n + 660) * a(n-1) - 16 * (n - 1) * (570*n^5 - 3570*n^4 + 8209*n^3 - 8451*n^2 + 3728*n - 540) * a(n-2) + 64 * (n - 2) * (n - 1) * (2*n - 5) * (57*n^3 - 129*n^2 + 76*n - 12) * a(n-3) for n > 2.

Prog

(PARI) a(n, s=2, t=1, u=2) = sum(k=0, n\s, binomial(t*n, k)*binomial(u*n, n-s*k));

Crossrefs

Cf. A005809, A370196, A395614.

Cf. A369439, A394709, A395612.

Sequence in context: A155609 A220542 A199213 * A002003 A123164 A264229

Adjacent sequences: A394704 A394705 A394706 * A394708 A394709 A394710

Keyword

nonn,easy

Author

Seiichi Manyama, May 01 2026

Status

approved