A116382 Riordan array (1/sqrt(1-4*x^2), (1-2*x^2*c(x^2))*(x^2*c(x^2))/(x*(1-x-x^2*c(x^2)))) where c(x) is the g.f. of A000108.

1, 0, 1, 2, 1, 1, 0, 3, 2, 1, 6, 4, 5, 3, 1, 0, 10, 10, 8, 4, 1, 20, 15, 21, 19, 12, 5, 1, 0, 35, 42, 42, 32, 17, 6, 1, 70, 56, 84, 92, 77, 50, 23, 7, 1, 0, 126, 168, 192, 180, 131, 74, 30, 8, 1, 252, 210, 330, 405, 400, 326, 210, 105, 38, 9, 1

Offset: 0

Author

Paul Barry, Feb 12 2006

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A116383. Diagonal sums are A116384.

First column has e.g.f. Bessel_I(0,2*x) (A000984 with interpolated zeros).

Second column has e.g.f. Bessel_I(1,2*x) + Bessel_I(2,2*x) (A037952).

Third column has e.g.f. Bessel_I(2,2*x) + 2*Bessel_I(3,2*x) + Bessel_I(4,2*x) (A116385).

A binomial-Bessel triangle: column k has e.g.f. Sum_{j=0..k} C(k,j) * Bessel_I(k+j,2*x).

Examples

			Triangle begins
    1;
    0,   1;
    2,   1,   1;
    0,   3,   2,   1;
    6,   4,   5,   3,   1;
    0,  10,  10,   8,   4,   1;
   20,  15,  21,  19,  12,   5,   1;
    0,  35,  42,  42,  32,  17,   6,   1;
   70,  56,  84,  92,  77,  50,  23,   7,  1;
    0, 126, 168, 192, 180, 131,  74,  30,  8, 1;
  252, 210, 330, 405, 400, 326, 210, 105, 38, 9, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Programs

GAP

Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j-> (-1)^(n-j)*Binomial(n,j)*Sum([0..j], m-> Binomial(j,m-k)*Binomial(m,j-m)  ))))); # G. C. Greubel, May 22 2019

Magma

T:= func< n,k | (&+[(-1)^(n-j)*Binomial(n,j)*(&+[Binomial(j,m-k)* Binomial(m,j-m): m in [0..j]]): j in [0..n]]) >;
[[T(n,k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 22 2019

Mathematica

T[n_, k_] := Sum[(-1)^(n-j)*Binomial[n, j]*Sum[Binomial[j, i-k]* Binomial[i, j-i], {i, 0, j}], {j, 0, n}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 24 2018 *)

PARI

{T(n,k) = sum(j=0,n, (-1)^(n-j)*binomial(n,j)*sum(m=0,j, binomial(j,m-k)*binomial(m,j-m) ))}; \\ G. C. Greubel, May 22 2019

Sage

def T(n, k): return sum((-1)^(n-j)*binomial(n,j)*sum(binomial(j,m-k)*binomial(m,j-m) for m in (0..j)) for j in (0..n))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 22 2019

Formula

Riordan array (1/sqrt(1-4*x^2), sqrt(1-4*x^2)*(1-sqrt(1-4*x^2))/(x-2*x^2 + x*sqrt(1-4*x^2))).

Number triangle T(n,k) = Sum{j=0..n} (-1)^(n-j)* C(n,j)*Sum_{i=0..j} C(j,i-k)*C(i,j-i).