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%I #13 Mar 08 2017 02:27:16
%S 1,1,1,3,2,1,7,9,3,1,19,28,18,4,1,51,95,70,30,5,1,141,306,285,140,45,
%T 6,1,393,987,1071,665,245,63,7,1,1107,3144,3948,2856,1330,392,84,8,1,
%U 3139,9963,14148,11844,6426,2394,588,108,9,1
%N A117852*A130595 as lower triangular matrices.
%C Mirror image of triangle in A135091.
%C Exponential Riordan array [exp(x)*Bessel_I(0,2*x), x] = A007318 * A109187. - _Peter Bala_, Feb 12 2017
%H G. C. Greubel, <a href="/A171128/b171128.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%F Sum_{k=0..n} T(n,k)*x^k = A126869(n), A002426(n), A000984(n), A026375(n), A081671(n), A098409(n), A098410(n), A104454(n) for x = -1,0,1,2,3,4,5,6 respectively.
%F T(n,k) = binomial(n,k)*A002426(n-k). - _Philippe Deléham_, Dec 12 2009
%F From _Peter Bala_, Feb 12 2017: (Start)
%F T(n,k) = Sum_{j = 0..floor((n-k)/2)} n!/((n-k-2*j)!*j!^2*k!).
%F T(n,k) = n/k*T(n-1,k-1) with T(n,0) = A002426(n).
%F (n - k)^2*T(n,k) = n*(2*n - 2*k - 1)*T(n-1,k) + 3*n*(n - 1)*T(n-2,k).
%F O.g.f. = 1/sqrt((1 - (1 + t)*z)^2 - 4*z^2) = 1 + (1 + t)*z + (3 + 2*t + t^2)*z^2 + (7 + 9*t + 3*t^2 + t^3 )*z^3 + ....
%F E.g.f. Bessel_I(0,2*x) * exp((1 + t)*x) = 1 + (1 + t)*z + (3 + 2*t + t^2)*z^2/2! + (7 + 9*t + 3*t^2 + t^3 )*z^3/3! + ....
%F n-th row polynomial R(n,t) = Sum_{k = 0..floor(n/2)} binomial(n,2*k)*binomial(2*k,k)*(1 + t)^(n-2*k) = coefficient of x^n in the expansion of (1 + (1 + t)*x + x^2)^n.
%F The polynomials R(n, t - 1) are the row polynomials of A109187.
%F d/dt(R(n,t)) = n*R(n-1,t).
%F Moment representation on a finite interval: R(n,t) = 1/Pi * Integral_{x = t-1 .. t+3} x^n/sqrt((t + 3 - x)*(x - t + 1)) dx.
%F The zeros of the row polynomials appear to lie on the vertical line Re(z) = -1 in the complex plane, and the zeros of R(n,t) and R(n+1,t) appear to interlace along this line.
%F (End)
%e Triangle begins:
%e 1
%e 1 1
%e 3 2 1
%e 7 9 3 1
%e 19 28 18 4 1
%e ...
%e From _Peter Bala_, Feb 12 2017: (Start)
%e The infinitesimal generator begins
%e 0
%e 1 0
%e 2 2 0
%e 0 6 3 0
%e -6 0 12 4 0
%e 0 -30 0 20 5 0
%e 80 0 -90 0 30 6 0
%e 0 560 0 -210 0 42 7 0
%e -2310 0 2240 0 -420 0 56 8 0
%e ....
%e and equals the generalized exponential Riordan array [x + log(Bessel_I(0,2*x), x], and so has integer entries. (End)
%t A002426[n_] := Sum[Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}]; Table[ Binomial[n, k]*A002426[n - k], {n, 0, 99}, {k, 0, n} ]//Flatten (* G_. C. Greubel_, Mar 07 2017 *)
%Y A000984 (row sums), A135091 (row reversed). Cf. A002426, A117852, A130595, A109187.
%K nonn,tabl
%O 0,4
%A _Philippe Deléham_, Dec 04 2009
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