A172991 Triangle of binomial sums read by rows: T(n,k) = sum(C(2n-2k-i,i) * C(2k-i,i), i=0..min(k,n-k)).

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 11, 6, 1, 1, 8, 22, 22, 8, 1, 1, 10, 37, 63, 37, 10, 1, 1, 12, 56, 136, 136, 56, 12, 1, 1, 14, 79, 249, 376, 249, 79, 14, 1, 1, 16, 106, 410, 849, 849, 410, 106, 16, 1, 1, 18, 137, 627, 1663, 2317, 1663, 627, 137, 18, 1, 1, 20, 172, 908, 2942, 5371, 5371, 2942, 908, 172, 20, 1, 1, 22, 211, 1261, 4826, 11017, 14545, 11017, 4826, 1261, 211, 22, 1

Offset: 0

Author

Emanuele Munarini, Apr 07 2011

Keywords

• nonn
• easy
• tabl

Comments

The matrix inverse starts

1;

-1,1;

1,-2,1;

-1,4,-4,1;

0,-8,13,-6,1;

7,12,-38,26,-8,1;

-35,-12,114,-101,43,-10,1; - R. J. Mathar, Mar 22 2013

Examples

			G.f. =
1 +
(y + 1)*x +
(y^2 + 2*y + 1)*x^2 +
(y^3 + 4*y^2 + 4*y + 1)*x^3 +
(y^4 + 6*y^3 + 11*y^2 + 6*y + 1)*x^4 + ...
Triangle begins:
1,
1,  1,
1,  2,  1,
1,  4,  4,   1,
1,  6, 11,   6,   1,
1,  8, 22,  22,   8,   1,
1, 10, 37,  63,  37,  10,  1,
1, 12, 56, 136, 136,  56, 12,  1,
1, 14, 79, 249, 376, 249, 79, 14,  1
		

Crossrefs

Cf. A188648, A054142, A027989.

Programs

Mathematica

Flatten[Table[Sum[Binomial[2n-2k-i,i]Binomial[2k-i,i],{i,0,Min[k,n-k]}],{n,0,12},{k,0,n}]]

Maxima

create_list(sum(binomial(2*n-2*k-i,i)*binomial(2*k-i,i),i,0,min(k,n-k)),n,0,10,k,0,n);

Formula

G.f.: (1 -x -x*y -2*x^2*y +x^3*y +x^3*y^2 +4*x^4*y^2 -x^6*y^3) / (1 -2*x +x^2 -2*x*y+2*x^3*y +x^2*y^2 +2*x^3*y^2 +3*x^4*y^2 -2*x^5*y^2 -2*x^5*y^3 -6*x^6*y^3 +x^8*y^4).

Central coefficients T(2n,n) = A188648.