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A171128 A117852*A130595 as lower triangular matrices. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Mirror image of triangle in A135091.

Exponential Riordan array [exp(x)*Bessel_I(0,2*x), x] = A007318 * A109187. - Peter Bala, Feb 12 2017

LINKS

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

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)

EXAMPLE

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)

MATHEMATICA

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 *)

CROSSREFS

A000984 (row sums), A135091 (row reversed). Cf. A002426, A117852, A130595, A109187.

Sequence in context: A059380 A145035 A192020 * A122832 A056151 A134436

Adjacent sequences:  A171125 A171126 A171127 * A171129 A171130 A171131

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Dec 04 2009

STATUS

approved

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Last modified April 13 06:52 EDT 2021. Contains 342935 sequences. (Running on oeis4.)