A137477 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.

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, -1212, 0, 5040, 0, 8, 0, -544, 0, 9696, 0, -40320, 1, 0, -140, 0, 6108, 0, -92304, 0, 362880, 0, -10, 0, 1400, 0, -61080, 0, 923040, 0, -3628800

Offset

1,3

Comments

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

Links

G. C. Greubel, Table of n, a(n) for the first 25 rows, flattened

Eric Weisstein's World of Mathematics, Spherical Bessel Differential Equation

Formula

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.

Example

Triangle begins with:
 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,  -1212,      0,   5040;
 0,   8,    0, -544,      0,   9696,      0, -40320;
 1,   0, -140,    0,   6108,      0, -92304,      0, 362880;

Mathematica

k = 1;
B[x, -1] = 0; B[x, 0] = 1;
B[x_, n_]:= B[x, n]= (-2/x)*B[x, n-1] -(k^2 -(n*(n-1)/x^2))*B[x, n-2];
Table[ExpandAll[B[x, n]/.x->1/y], {n, 0, 10}] (* polynomials *)
Table[CoefficientList[B[x, n] /. x -> 1/y, y], {n, 0, 10}]//Flatten
Table[Apply[Plus, CoefficientList[B[x, n] /. x -> 1/y, y]], {n, 0, 10}] (* row sums *)

Crossrefs

Sequence in context: A241218 A266904 A299198 * A181297 A196776 A336345

Adjacent sequences: A137474 A137475 A137476 * A137478 A137479 A137480

Keyword

tabl,sign

Author

Roger L. Bagula, Apr 21 2008

Status

approved