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

0,4

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).

Links

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

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).

Example

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;

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 *)

Prog

(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
(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
(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
(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

Crossrefs

Sequence in context: A288166 A126258 A235501 * A050606 A352521 A277721

Adjacent sequences: A116379 A116380 A116381 * A116383 A116384 A116385

Keyword

easy,nonn,tabl

Author

Paul Barry, Feb 12 2006

Status

approved