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A182013 Triangle of partial sums of Motzkin numbers. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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.

LINKS

Table of n, a(n) for n=0..56.

FORMULA

T(n,k) = sum(M(i),i=k..n), 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.

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}]]

PROG

(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);

CROSSREFS

Cf. A001006, A086615, A097861, A182015, A182016, A182017.

Sequence in context: A204922 A057669 A243610 * A144333 A126136 A140169

Adjacent sequences:  A182010 A182011 A182012 * A182014 A182015 A182016

KEYWORD

nonn,tabl

AUTHOR

Emanuele Munarini, Apr 06 2012

STATUS

approved

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Last modified April 20 14:27 EDT 2019. Contains 322310 sequences. (Running on oeis4.)