A171128 A117852*A130595 as lower triangular matrices.
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, 6, 1, 393, 987, 1071, 665, 245, 63, 7, 1, 1107, 3144, 3948, 2856, 1330, 392, 84, 8, 1, 3139, 9963, 14148, 11844, 6426, 2394, 588, 108, 9, 1
Offset: 0
Examples
Triangle begins:
1
1 1
3 2 1
7 9 3 1
19 28 18 4 1
...
From _Peter Bala_, Feb 12 2017: (Start)
The infinitesimal generator begins
0
1 0
2 2 0
0 6 3 0
-6 0 12 4 0
0 -30 0 20 5 0
80 0 -90 0 30 6 0
0 560 0 -210 0 42 7 0
-2310 0 2240 0 -420 0 56 8 0
....
and equals the generalized exponential Riordan array [x + log(Bessel_I(0,2*x)), x], and so has integer entries. (End)
Links
- G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Programs
Formula
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.
T(n,k) = binomial(n,k)*A002426(n-k). - Philippe Deléham, Dec 12 2009
From Peter Bala, Feb 12 2017: (Start)
T(n,k) = Sum_{j = 0..floor((n-k)/2)} n!/((n-k-2*j)!*j!^2*k!).
T(n,k) = n/k*T(n-1,k-1) with T(n,0) = A002426(n).
(n - k)^2*T(n,k) = n*(2*n - 2*k - 1)*T(n-1,k) + 3*n*(n - 1)*T(n-2,k).
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 + ....
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! + ....
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.
The polynomials R(n, t - 1) are the row polynomials of A109187.
d/dt(R(n,t)) = n*R(n-1,t).
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.
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.
(End)
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