A259554 a(n) = Sum_{i=0..n} (2^(i)*(-1)^(i+n)*C(n,i)*C(2*n+i-1,n-1)).
1, 7, 52, 403, 3206, 25954, 212738, 1760035, 14666470, 122920642, 1035046816, 8749594462, 74207078908, 631140253072, 5381022869822, 45975731083555, 393556869530630, 3374504760608026, 28977403637496104, 249167023897718138
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- 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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Maple
a := n -> n*hypergeom([2*n+1, -n+1], [2], -1): seq(simplify(a(n)), n=1..9); # Peter Luschny, Oct 07 2016
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Mathematica
Table[Sum[2^i * (-1)^(i+n) * Binomial[n, i] * Binomial[2*n+i-1, n-1], {i, 0, n}], {n,1,20}] (* Vaclav Kotesovec, Jul 01 2015 *) -
Maxima
a(n):=sum(2^(i)*(-1)^(i+n)*binomial(n,i)*binomial(2*n+i-1,n-1),i,0,n);
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PARI
a(n) = sum(i=0, n, 2^i*(-1)^(i+n)*binomial(n, i)*binomial(2*n+i-1, n-1)); \\ Michel Marcus, Jul 02 2015
Formula
G.f.: A(x) = x*B(x)'/B(x), where B(x) is g.f. of A003168.
Recurrence: 4*n*(2*n-1)*(17*n^2 - 51*n + 38)*a(n) = (1207*n^4 - 4828*n^3 + 6659*n^2 - 3662*n + 672)*a(n-1) - 2*(n-2)*(2*n-3)*(17*n^2 - 17*n + 4)*a(n-2). - Vaclav Kotesovec, Jul 01 2015
a(n) ~ (71+17*sqrt(17))^n / (17^(1/4) * sqrt(Pi*n) * 2^(4*n+1)). - Vaclav Kotesovec, Jul 01 2015
a(n) = (1/2)*Sum_{k = 0..n} binomial(n-1,n-k)*binomial(2*n+k-1,k). - Vladimir Kruchinin, Oct 07 2016
a(n) = n*hypergeom([2*n+1, -n+1], [2], -1). - Peter Luschny, Oct 07 2016
From Peter Bala, Nov 08 2022: (Start)
a(n) = (1/2)*[x^n] ( (1 - x)/(1 - 2*x) )^(2*n). Cf. A002002(n) = [x^n] ( (1 - x)/(1 - 2*x) )^n.
a(n) = (1/2)*Sum_{k = 0..n} (-1)^(n-k)*2^k*binomial(2*n,n-k)*binomial(2*n+k-1,k).
a(n) = (1/2)*(-1)^n*binomial(2*n,n)*hypergeom( [-n, 2*n], [n+1], 2).
The Gauss congruences hold: a(n*p^r) == a(n^p^(r-1)) (mod p^r) for all primes p >= 3 and all positive integers n and r. (End)