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A182013 Triangle of partial sums of Motzkin numbers.

%I A182013 #11 Oct 31 2021 03:06:28
%S A182013 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 A182013 72,51,216,215,214,212,208,199,178,127,539,538,537,535,531,522,501,
%U A182013 450,323,1374,1373,1372,1370,1366,1357,1336,1285,1158,835,3562,3561
%N A182013 Triangle of partial sums of Motzkin numbers.
%F A182013 T(n, k) = Sum_{i=k..n} M(i), where the M(n)'s are the Motzkin numbers.
%F A182013 Recurrence: T(n+1, k+1) = T(n, k) + M(n+1) - M(k).
%F A182013 G.f. (M(x) - y*M(x*y))/((1 - x)*(1 - y)), where M(x) is the generating series for Motzkin numbers.
%e A182013 Triangle begins:
%e A182013   1
%e A182013   2,   1
%e A182013   4,   3,   2
%e A182013   8,   7,   6,   4
%e A182013   17,  16,  15,  13,  9
%e A182013   38,  37,  36,  34,  30,  21
%e A182013   89,  88,  87,  85,  81,  72,  51
%e A182013   216, 215, 214, 212, 208, 199, 178, 127
%e A182013   539, 538, 537, 535, 531, 522, 501, 450, 323
%t A182013 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 A182013 (Maxima) M(n):=coeff(expand((1+x+x^2)^(n+1)),x^n)/(n+1);
%o A182013 create_list(sum(M(i),i,k,n),n,0,6,k,0,n);
%Y A182013 Diagonal elements = Motzkin numbers (A001006).
%Y A182013 First column = partial sums of Motzkin numbers (A086615).
%Y A182013 Row sums = A097861(n+1).
%Y A182013 Diagonal sums = A182015.
%Y A182013 Row square-sums = A182017.
%Y A182013 Central coefficients = A182016.
%K A182013 nonn,tabl
%O A182013 0,2
%A A182013 _Emanuele Munarini_, Apr 06 2012