A201639 Triangle read by rows, T(n,k) for 0<=k<=n, generalizes the Motzkin lattice paths with weights of A003645.

1, 4, 1, 20, 8, 1, 112, 56, 12, 1, 672, 384, 108, 16, 1, 4224, 2640, 880, 176, 20, 1, 27456, 18304, 6864, 1664, 260, 24, 1, 183040, 128128, 52416, 14560, 2800, 360, 28, 1, 1244672, 905216, 396032, 121856, 27200, 4352, 476, 32, 1, 8599552, 6449664, 2976768

Offset: 0

Author

Peter Luschny, Sep 20 2012

Keywords

• nonn
• tabl

Examples

			[0] [1]
[1] [4, 1]
[2] [20, 8, 1]
[3] [112, 56, 12, 1]
[4] [672, 384, 108, 16, 1]
[5] [4224, 2640, 880, 176, 20, 1]
[6] [27456, 18304, 6864, 1664, 260, 24, 1]
[7] [183040, 128128, 52416, 14560, 2800, 360, 28, 1]
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..9044

Crossrefs

Sum of row n is A194723(n+1).

Cf. A003645.

Programs

GAP

Flat(List([0..10], n->List([0..n],k->(k+1)*2^(n-k)*Binomial(2*(n+1),n-k)/(n+1)))); # Muniru A Asiru, Apr 07 2018

Magma

/* As triangle */ [[(k+1)*2^(n-k)*Binomial(2*(n+1),n-k)/(n+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Apr 07 2018

Mathematica

Flatten[Table[(k + 1) 2^(n - k) Binomial[2 (n + 1), n - k] / (n + 1), {n, 0, 11}, {k, 0, n}]] (* Vincenzo Librandi, Apr 07 2018 *)

PARI

T(n,k) = (k+1)*2^(n-k)*binomial(2*(n+1),n-k)/(n+1);
tabl(nn) = for(n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Apr 07 2018

Sage

def A201639_triangle(dim):
    T = matrix(ZZ,dim,dim)
    for n in range(dim): T[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            T[n,k] = T[n-1,k-1]+4*T[n-1,k]+4*T[n-1,k+1]
    return T
A201639_triangle(9)

Formula

Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n>=1 T(n,k) = T(n-1,k-1)+4*T(n-1,k)+4*T(n-1,k+1).

G.f.: -(4*x+sqrt(1-8*x)-1)/((4*x^2-x)*y+sqrt(1-8*x)*x*y+8*x^2). - Vladimir Kruchinin, Apr 06 2018

T(n,k) = (k+1)*2^(n-k)*C(2*(n+1),n-k)/(n+1). - Vladimir Kruchinin, Apr 06 2018