A190252 Riordan matrix (g(x),x*g(x)), where g(x) = (1-x-sqrt(1-2*x-3*x^2-4*x^3))/(2*x^2*(1+x)).

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 12, 14, 9, 4, 1, 31, 38, 28, 14, 5, 1, 83, 106, 84, 48, 20, 6, 1, 227, 301, 252, 157, 75, 27, 7, 1, 634, 864, 758, 504, 265, 110, 35, 8, 1, 1799, 2508, 2283, 1602, 906, 417, 154, 44, 9, 1, 5171, 7348, 6897, 5056, 3035, 1512, 623, 208, 54, 10, 1, 15027, 21699, 20903, 15894, 10020, 5324, 2387, 894, 273, 65, 11, 1

Offset

0,4

Comments

First column = A071359(n+1).

Central coefficients = A190253.

Row sums = A190254.

Diagonal sums = A190255.

Links

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

Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.

Formula

T(n,k) = [x^(n-k)]g(x)^(k+1), where g(x) = (1-x-sqrt(1-2*x-3*x^2-4*x^3)) / (2*x^2*(1+x)).

T(n,k) = sum(i=0..(n-k)/2, binomial(2*i+k,i)*(k+1)/(i+k+1) * sum(j=0..i, binomial(i,j)*binomial(n-j,2*i+k) ) ).

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

Example

Triangle begins:
    1;
    1,   1;
    2,   2,   1;
    5,   5,   3,   1;
   12,  14,   9,   4,   1;
   31,  38,  28,  14,   5,   1;
   83, 106,  84,  48,  20,   6,   1;
  227, 301, 252, 157,  75,  27,   7,   1;
  634, 864, 758, 504, 265, 110,  35,   8,   1;
  ...

Mathematica

Flatten[Table[Sum[Binomial[2i+k, i](k+1)/(i+k+1) Sum[Binomial[i, j] Binomial[n-j, 2i+k], {j, 0, i}], {i, 0, (n-k)/2}], {n, 0, 12}, {k, 0, n}]]

Prog

(Maxima) create_list(sum(binomial(2*i+k, i)*(k+1)/(i+k+1)*sum( binomial(i, j)*binomial(n-j, 2*i+k), j, 0, i), i, 0, (n-k)/2), n, 0, 12, k, 0, n);

Crossrefs

Cf. A190253, A190254, A190255.

Sequence in context: A178518 A299499 A190215 * A141751 A079222 A033184

Adjacent sequences: A190249 A190250 A190251 * A190253 A190254 A190255

Keyword

nonn,tabl

Author

Emanuele Munarini, May 06 2011

Status

approved