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
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
Programs
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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.