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%I #11 Oct 31 2021 03:06:28
%S 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,
%T 72,51,216,215,214,212,208,199,178,127,539,538,537,535,531,522,501,
%U 450,323,1374,1373,1372,1370,1366,1357,1336,1285,1158,835,3562,3561
%N Triangle of partial sums of Motzkin numbers.
%F T(n, k) = Sum_{i=k..n} M(i), where the M(n)'s are the Motzkin numbers.
%F Recurrence: T(n+1, k+1) = T(n, k) + M(n+1) - M(k).
%F G.f. (M(x) - y*M(x*y))/((1 - x)*(1 - y)), where M(x) is the generating series for Motzkin numbers.
%e Triangle begins:
%e 1
%e 2, 1
%e 4, 3, 2
%e 8, 7, 6, 4
%e 17, 16, 15, 13, 9
%e 38, 37, 36, 34, 30, 21
%e 89, 88, 87, 85, 81, 72, 51
%e 216, 215, 214, 212, 208, 199, 178, 127
%e 539, 538, 537, 535, 531, 522, 501, 450, 323
%t 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}]]
%o (Maxima) M(n):=coeff(expand((1+x+x^2)^(n+1)),x^n)/(n+1);
%o create_list(sum(M(i),i,k,n),n,0,6,k,0,n);
%Y Diagonal elements = Motzkin numbers (A001006).
%Y First column = partial sums of Motzkin numbers (A086615).
%Y Row sums = A097861(n+1).
%Y Diagonal sums = A182015.
%Y Row square-sums = A182017.
%Y Central coefficients = A182016.
%K nonn,tabl
%O 0,2
%A _Emanuele Munarini_, Apr 06 2012
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