OFFSET
0,4
COMMENTS
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
KEYWORD
AUTHOR
Paul Barry, Feb 12 2006
STATUS
approved