A190215 Riordan matrix ((1-x-x^2)/(1-2x-x^2),(x-x^2-x^3)/(1-2x-x^2)).

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 12, 14, 9, 4, 1, 29, 38, 28, 14, 5, 1, 70, 102, 84, 48, 20, 6, 1, 169, 271, 246, 157, 75, 27, 7, 1, 408, 714, 707, 496, 265, 110, 35, 8, 1, 985, 1868, 2001, 1526, 896, 417, 154, 44, 9, 1, 2378, 4858, 5592, 4596, 2930, 1500, 623, 208, 54, 10, 1, 5741, 12569, 15461, 13602, 9330, 5186, 2373, 894, 273, 65, 11, 1

Offset

0,4

Comments

Row sums = A052963.

Diagonal sums = A052960.

Central coefficients = A190315.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

T(n,k) = Sum_{i=0..n-k} (binomial(i+k,k)*Sum_{j=0..n-k-i} (binomial(i+j-1,j)*binomial(j,n-k-i-j) )).

Recurrence: T(n+3,k+1) = 2 T(n+2,k+1) + T(n+2,k) + T(n+1,k+1) - T(n+1,k) - T(n,k).

Example

Triangle begins:
    1;
    1,   1;
    2,   2,   1;
    5,   5,   3,   1;
   12,  14,   9,   4,   1;
   29,  38,  28,  14,   5,   1;
   70, 102,  84,  48,  20,   6,   1;
  169, 271, 246, 157,  75,  27,   7,   1;
  408, 714, 707, 496, 265, 110,  35,   8,   1;

Mathematica

Flatten[Table[Sum[Binomial[i+k, k]Sum[Binomial[i+j-1, j]Binomial[j, n-k-i-j], {j, 0, n-k-i}], {i, 0, n-k}], {n, 0, 12}, {k, 0, n}]]

Prog

(Maxima) create_list(sum(binomial(i+k, k)*sum(binomial(i+j-1, j)*binomial(j, n-k-i-j), j, 0, n-k-i), i, 0, n-k), n, 0, 12, k, 0, n);
(PARI) for(n=0, 10, for(k=0, n, print1(sum(j=0, n-k, binomial(j+k, k)* sum(r=0, n-k-j, binomial(j+r-1, r)*binomial(r, n-k-j-r))), ", "))) \\ G. C. Greubel, Dec 27 2017

Crossrefs

Cf. A052963, A052960, A190315.

Sequence in context: A333724 A178518 A299499 * A190252 A141751 A079222

Adjacent sequences: A190212 A190213 A190214 * A190216 A190217 A190218

Keyword

nonn,tabl

Author

Emanuele Munarini, May 10 2011

Status

approved