A182013 Triangle of partial sums of Motzkin numbers.

1, 2, 1, 4, 3, 2, 8, 7, 6, 4, 17, 16, 15, 13, 9, 38, 37, 36, 34, 30, 21, 89, 88, 87, 85, 81, 72, 51, 216, 215, 214, 212, 208, 199, 178, 127, 539, 538, 537, 535, 531, 522, 501, 450, 323, 1374, 1373, 1372, 1370, 1366, 1357, 1336, 1285, 1158, 835, 3562, 3561

Offset: 0

Author

Emanuele Munarini, Apr 06 2012

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1
  2,   1
  4,   3,   2
  8,   7,   6,   4
  17,  16,  15,  13,  9
  38,  37,  36,  34,  30,  21
  89,  88,  87,  85,  81,  72,  51
  216, 215, 214, 212, 208, 199, 178, 127
  539, 538, 537, 535, 531, 522, 501, 450, 323
		

Crossrefs

Diagonal elements = Motzkin numbers (A001006).

First column = partial sums of Motzkin numbers (A086615).

Row sums = A097861(n+1).

Diagonal sums = A182015.

Row square-sums = A182017.

Central coefficients = A182016.

Programs

Mathematica

M[n_] := If[n==0, 1, Coefficient[(1+x+x^2)^(n+1), x^n]/(n+1)]; Flatten[Table[Sum[M[i], {i,k,n}], {n,0,30}, {k,0,n}]]

Maxima

M(n):=coeff(expand((1+x+x^2)^(n+1)),x^n)/(n+1);
create_list(sum(M(i),i,k,n),n,0,6,k,0,n);

Formula

T(n, k) = Sum_{i=k..n} M(i), where the M(n)'s are the Motzkin numbers.

Recurrence: T(n+1, k+1) = T(n, k) + M(n+1) - M(k).

G.f. (M(x) - y*M(x*y))/((1 - x)*(1 - y)), where M(x) is the generating series for Motzkin numbers.