A137477 - OEIS

%I #20 Nov 28 2024 13:27:32

%S 1,0,-2,-1,0,6,0,4,0,-24,1,0,-26,0,120,0,-6,0,156,0,-720,-1,0,68,0,

%T -1212,0,5040,0,8,0,-544,0,9696,0,-40320,1,0,-140,0,6108,0,-92304,0,

%U 362880,0,-10,0,1400,0,-61080,0,923040,0,-3628800

%N A triangular sequence of coefficients from the inverse substitution of the spherical Bessel polynomial recursion: B(x, n) = (-2/x)*B(x, n-1) - (k^2 - (n*(n-1)/x^2))*B(x, n-2), with k=1 and substitution x->1/y.

%C Row sums are {1, -2, 5, -20, 95, -570, 3895, -31160, 276545, -2765450, 30143405, ...}.

%H G. C. Greubel, <a href="/A137477/b137477.txt">Table of n, a(n) for the first 25 rows, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SphericalBesselDifferentialEquation.html">Spherical Bessel Differential Equation</a>

%F B(x, n) = (-2/x)*B(x, n-1) - (k^2 - (n*(n-1)/x^2))*B(x, n-2), with k=1 and the substitution x -> 1/y.

%e Triangle begins with:

%e  1;

%e  0,  -2;

%e -1,   0,    6;

%e  0,   4,    0,  -24;

%e  1,   0,  -26,    0,    120;

%e  0,  -6,    0,  156,      0,   -720;

%e -1,   0,   68,    0,  -1212,      0,   5040;

%e  0,   8,    0, -544,      0,   9696,      0, -40320;

%e  1,   0, -140,    0,   6108,      0, -92304,      0, 362880;

%t k = 1;

%t B[x, -1] = 0; B[x, 0] = 1;

%t B[x_, n_]:= B[x, n]= (-2/x)*B[x, n-1] -(k^2 -(n*(n-1)/x^2))*B[x, n-2];

%t Table[ExpandAll[B[x, n]/.x->1/y], {n, 0, 10}] (* polynomials *)

%t Table[CoefficientList[B[x, n] /. x -> 1/y, y], {n, 0, 10}]//Flatten

%t Table[Apply[Plus, CoefficientList[B[x, n] /. x -> 1/y, y]], {n, 0, 10}] (* row sums *)

%K tabl,sign

%O 1,3

%A _Roger L. Bagula_, Apr 21 2008