A069835 Define an array as follows: b(i,0)=b(0,j)=1, b(i,j) = 2*b(i-1,j-1) + b(i-1,j) + b(i,j-1). Then a(n) = b(n,n).

1, 4, 22, 136, 886, 5944, 40636, 281488, 1968934, 13875544, 98365972, 700701808, 5011371964, 35961808432, 258805997752, 1867175631136, 13500088649734, 97794850668952, 709626281415076, 5157024231645616, 37528209137458516, 273431636191026064, 1994448720786816712

Offset: 0

Author

Benoit Cloitre, May 03 2002

Keywords

• nonn
• easy

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

Examples

			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, ...
		

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.

Crossrefs

Cf. A001850.

Programs

GAP

List([0..25],n->Sum([0..n],k->Binomial(n,k)^2*3^k)); # Muniru A Asiru, Jul 29 2018

Haskell

a069835 n = a081577 (2 * n) n -- Reinhard Zumkeller, Mar 16 2014

Mathematica

Table[Hypergeometric2F1[-n, -n, 1, 3], {n, 0, 21}] (* Arkadiusz Wesolowski, Aug 13 2012 *)

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)<Charles R Greathouse IV, Mar 18 2017

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