A378405 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n+k,n-2*k).
1, 1, 3, 13, 47, 171, 651, 2507, 9703, 37831, 148393, 584673, 2312267, 9174179, 36500257, 145566333, 581746503, 2329206823, 9341025429, 37516150599, 150874376997, 607479424817, 2448608334087, 9879562243961, 39897969991075, 161260133795371
Offset: 0
Keywords
Programs
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Mathematica
Table[Sum[Binomial[n, k]*Binomial[n + k, n - 2*k], {k, 0, n/2}], {n, 0, 30}] (* Vaclav Kotesovec, Nov 25 2024 *) -
PARI
a(n) = sum(k=0, n\2, binomial(n, k)*binomial(n+k, n-2*k));
Formula
a(n) = [x^n] (1 + x + x^2 * (1 + x)^2)^n.
a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 4.1236218756427610008124277125077732535524468472302771364162049292... is the greatest root of the equation 31 - 256*d + 30*d^2 - 104*d^3 + 27*d^4 = 0 and c = 0.3580097056143148810957018324419771260252367819271313726816... is the positive real root of the equation -31 - 3024*c + 51376*c^2 - 257536*c^3 + 386304*c^4 = 0. - Vaclav Kotesovec, Nov 25 2024