A152440 Riordan matrix (1/(1-x-x^2),x/(1-x-x^2)^2).

1, 1, 1, 2, 3, 1, 3, 9, 5, 1, 5, 22, 20, 7, 1, 8, 51, 65, 35, 9, 1, 13, 111, 190, 140, 54, 11, 1, 21, 233, 511, 490, 255, 77, 13, 1, 34, 474, 1295, 1554, 1035, 418, 104, 15, 1, 55, 942, 3130, 4578, 3762, 1925, 637, 135, 17, 1, 89, 1836, 7285, 12720, 12573, 7865, 3276

Offset: 0

Author

Emanuele Munarini, Dec 04 2008, Dec 05 2008

Keywords

• nonn
• tabl
• easy

Comments

From Philippe Deléham, Feb 20 2014: (Start)

T(n,0) = A000045(n+1);

T(n+1,1) = A001628(n);

T(n+2,2) = A001873(n);

T(n+3,3) = A001875(n).

Row sums are A238236(n). (End)

Examples

			Triangle begins:
1;
1, 1;
2, 3, 1;
3, 9, 5, 1;
5, 22, 20, 7, 1;
8, 51, 65, 35, 9, 1;
13, 111, 190, 140, 54, 11, 1;
21, 233, 511, 490, 255, 77, 13, 1, etc.
- _Philippe Deléham_, Feb 20 2014
		

Crossrefs

The first row is given by A000045.

Cf. A037027, A238241.

Formula

a(n,k) = sum( binomial(n-j-k,2k) binomial(n-j-k,j), j=0...(n-k)/2 )

a(n,k) = sum( binomial(i+2k,2k) binomial(n-i+k,i+2k), i=0...(n - k)/2 )

Recurrence: a(n+4,k+1) - 2 a(n+3,k+1) - a(n+3,k) - a(n+2,k+1) + 2 a(n+1,k+1) + a(n,k+1) = 0

GF for columns: 1/(1-x-x^2)(x/(1-x-x^2)^2)^k

GF: (1-x-x^2)/((1-x-x^2)^2-xy)

T(n,k) = A037027(n+k, 2*k). - Philippe Deléham, Feb 20 2014