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%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
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