A259557 a(n) = binomial(4*n-1, 2*n).
1, 3, 35, 462, 6435, 92378, 1352078, 20058300, 300540195, 4537567650, 68923264410, 1052049481860, 16123801841550, 247959266474052, 3824345300380220, 59132290782430712, 916312070471295267, 14226520737620288370
Offset: 0
Links
- V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.
Programs
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Magma
[Binomial(4*n-1, 2*n): n in [0..20]]; // Vincenzo Librandi, Jul 01 2015
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Mathematica
Table[Binomial[4 n - 1, 2 n], {n, 0, 30}] (* Vincenzo Librandi, Jul 01 2015 *) -
PARI
vector(20, n, n--; binomial(4*n-1, 2*n)) \\ Michel Marcus, Jul 01 2015
Formula
G.f. A(x)=1+x*B(x)'/B(x), where B(x) is g.f. of A079489.
a(n) = A100033(n-1) for n>0.
D-finite with recurrence n*(2*n-1)*a(n) -2*(4*n-1)*(4*n-3)*a(n-1)=0. - R. J. Mathar, Jul 06 2015
a(n) = [x^(2*n)] 1/(1 - x)^(2*n). - Ilya Gutkovskiy, Oct 10 2017
From Peter Bala, Jun 11 2023: (Start)
a(n) = (1/2) * [x^n] ( (1 + x)^2/( 1 - x) )^(2*n) for n >= 1.
Right-hand side of the identity (1/2)*Sum_{k = 0..n} binomial(4*n,k)*binomial(3*n-k-1,n-k) = binomial(4*n-1,2*n) for n >= 1.
a(n) = [x^n] E(x)^n, where E(x) = exp( Sum_{k >= 1} A119259(k)*x^k/k ). (End)
a(n) = Sum_{k = 0..2*n} (-1)^k*binomial(-n, k)*binomial(-3*n-k, 2*n-k) = Sum_{k = 0..2*n} (-1)^k*binomial(n+k-1, k)*binomial(5*n-1, 2*n-k). - Peter Bala, Jun 08 2024
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