A116382 - OEIS

%I #17 Dec 29 2023 12:55:33

%S 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,

%T 42,42,32,17,6,1,70,56,84,92,77,50,23,7,1,0,126,168,192,180,131,74,30,

%U 8,1,252,210,330,405,400,326,210,105,38,9,1

%N 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.

%C Row sums are A116383. Diagonal sums are A116384.

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

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

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

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

%H G. C. Greubel, <a href="/A116382/b116382.txt">Rows n = 0..100 of triangle, flattened</a>

%F 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))).

%F 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).

%e Triangle begins

%e     1;

%e     0,   1;

%e     2,   1,   1;

%e     0,   3,   2,   1;

%e     6,   4,   5,   3,   1;

%e     0,  10,  10,   8,   4,   1;

%e    20,  15,  21,  19,  12,   5,   1;

%e     0,  35,  42,  42,  32,  17,   6,   1;

%e    70,  56,  84,  92,  77,  50,  23,   7,  1;

%e     0, 126, 168, 192, 180, 131,  74,  30,  8, 1;

%e   252, 210, 330, 405, 400, 326, 210, 105, 38, 9, 1;

%t 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}];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 24 2018 *)

%o (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

%o (Magma)

%o 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]]) >;

%o [[T(n,k): k in [0..n]]: n in [0..10]] // _G. C. Greubel_, May 22 2019

%o (Sage)

%o 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))

%o [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, May 22 2019

%o (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

%K easy,nonn,tabl

%O 0,4

%A _Paul Barry_, Feb 12 2006