A136668 Triangle read by rows: coefficients of a Bessel polynomial recursion: P(x, n) = 2*(n-1)*P(x, n - 1)/x - n*P(x, n - 2) with substitution x -> 1/y.

1, 0, 1, -2, 0, 2, 0, -11, 0, 8, 8, 0, -74, 0, 48, 0, 119, 0, -632, 0, 384, -48, 0, 1634, 0, -6608, 0, 3840, 0, -1409, 0, 24032, 0, -81984, 0, 46080, 384, 0, -32798, 0, 389312, 0, -1178496, 0, 645120, 0, 18825, 0, -741056, 0, 6966848, 0, -19270656, 0, 10321920

Offset: 1

Author

Roger L. Bagula, Apr 03 2008

Keywords

• tabl
• sign

Comments

Row sums: {1, 1, 0, -3, -18, -129, -1182, -13281, -176478, -2704119, -46909362, ...}.

Examples

			Triangle begins as:
    1;
    0,     1;
   -2,     0,    2;
    0,   -11,    0,     8;
    8,     0,  -74,     0,    48;
    0,   119,    0,  -632,     0,    384;
  -48,     0, 1634,     0, -6608,      0, 3840;
    0, -1409,    0, 24032,     0, -81984,    0, 46080;
    ....
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1972, 10th edition, (and various reprintings), p. 631.

Links

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

Crossrefs

Cf. A106174, A123956.

Programs

Mathematica

P[x, 0]= 1; P[x, 1]= 1/x;
P[x_, n_]:= P[x, n] = 2*(n-1)*P[x, n-1]/x - n*P[x, n-2];
Table[ExpandAll[P[x, n] /. x -> 1/y], {n, 0, 10}];
Table[CoefficientList[P[x, n] /. x -> 1/y, y], {n, 0, 10}]//Flatten

Formula

P(x,0) = 1; P(x,1) = 1/x; P(x, n) = 2*(n-1)*P(x, n - 1)/x - n*P(x, n - 2); with substitution of x to 1/y.