OFFSET
0,2
COMMENTS
2^n*LegendreP(n,k) yields the central coefficients of (1 + 2*k*x + (k^2-1)*x^2)^n, with g.f. 1/sqrt(1 - 4*k*x + 4*x^2) and e.g.f. exp(2*k*x)BesselI(0, 2*sqrt(k^2-1)*x). - Paul Barry, May 25 2005
Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where D can have two colors. - Paul Barry, May 25 2005
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps can have three colors and H steps can have four colors. - N-E. Fahssi, Mar 31 2008
Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and two kinds of steps (1,1). - Joerg Arndt, Jul 01 2011
Hankel transform is 2^n*3^C(n+1,2) = (-1)^C(n+1,2)*A127946(n). - Paul Barry, Jan 24 2011
Central terms of triangle A152842. - Reinhard Zumkeller, May 01 2014
Diagonal of rational functions 1/(1 - x - y - 2*x*y), 1/(1 - x - y*z - 2*x*y*z). - Gheorghe Coserea, Jul 06 2018
The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - Peter Bala, Jan 07 2022
REFERENCES
Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.
LINKS
Arkadiusz Wesolowski, Table of n, a(n) for n = 0..250
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - N. J. A. Sloane, Oct 08 2012
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
FORMULA
From Vladeta Jovovic, May 13 2003: (Start)
a(n) = 2^n*LegendreP(n, 2) = 2^n*hypergeom([ -n, n+1], [1], -1/2) = 2^n*GegenbauerC(n, 1/2, 2) = Sum_{k=0..n} 3^k*binomial(n, k)^2.
D-finite with recurrence: a(n) = 4*(2*n-1)/n*a(n-1) - 4*(n-1)/n*a(n-2).
G.f.: 1/sqrt(1 - 8*x + 4*x^2). (End)
a(n) equals the central coefficient of (1 + 4*x + 3*x^2)^n. - Paul D. Hanna, Jun 03 2003
E.g.f.: exp(4*x)*Bessel_I(0, 2*sqrt(3)*x). - Paul Barry, Sep 20 2004
a(n) = Sum_{k=0..floor(n/2)} C(n, k)*C(2*(n-k), n)*(-1)^k*2^(n-2*k). - Paul Barry, May 25 2005
a(n) = Sum_{k=0..n} C(n, k)*C(n+k, k)*2^(n-k). - Paul Barry, May 25 2005
a(n) = Sum_{k=0..n} C(n, k)^2*3^k. - Paul Barry, Oct 15 2005
G.f.: 1/(1-4x-6x^2/(1-4x-3x^2/(1-4x-3x^2/(1-4x-3x^2/(1-... (continued fraction). - Paul Barry Jan 24 2011
Asymptotic: a(n) ~ sqrt(1/2 + 1/sqrt(3))*(1+sqrt(3))^(2*n)/sqrt(Pi*n). - Vaclav Kotesovec, Sep 11 2012
0 = a(n)*(16*a(n+1) - 48*a(n+2) + 8*a(n+3)) + a(n+1)*(-16*a(n+1) + 64*a(n+2) - 12*a(n+3)) + a(n+2)*(-4*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, Apr 21 2020
EXAMPLE
The array b is a rewriting of A081577:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, ...
1, 7, 22, 46, 79, 121, 172, 232, 301, 379, 466, ...
1, 10, 46, 136, 307, 586, 1000, 1576, 2341, 3322, 4546, ...
1, 13, 79, 307, 886, 2086, 4258, 7834, 13327, 21331, 32521, ...
MATHEMATICA
Table[Hypergeometric2F1[-n, -n, 1, 3], {n, 0, 21}] (* Arkadiusz Wesolowski, Aug 13 2012 *)
PROG
(PARI) a(n)=sum(k=0, n, binomial(n, k)^2*3^k)
(PARI) a(n)=if(n<0, 0, polcoeff((1+4*x+3*x^2)^n, n))
(PARI) /* as lattice paths: same as in A092566 but use */
steps=[[1, 0], [0, 1], [1, 1], [1, 1]]; /* note the double [1, 1] */
\\ Joerg Arndt, Jul 01 2011
(PARI) a(n)=pollegendre(n, 2)<<n \\ Charles R Greathouse IV, Mar 18 2017
(Haskell)
a069835 n = a081577 (2 * n) n -- Reinhard Zumkeller, Mar 16 2014
(GAP) List([0..25], n->Sum([0..n], k->Binomial(n, k)^2*3^k)); # Muniru A Asiru, Jul 29 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, May 03 2002
STATUS
approved