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

0,2

Links

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

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

Example

[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]

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 *)

Prog

(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)
(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
(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
(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

Crossrefs

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

Cf. A003645.

Sequence in context: A117380 A185420 A167432 * A078939 A135891 A049459

Adjacent sequences: A201636 A201637 A201638 * A201640 A201641 A201642

Keyword

nonn,tabl

Author

Peter Luschny, Sep 20 2012

Status

approved