A136533 Coefficients of Laguerre recursive polynomials with an (n+2)!/2 multiplication factor and alpha=a0 =0 from Hochstadt: P(x, n) = (2*n + a0 + 1 - x)*P(x, n - 1)/(n + 1) - n*P(x, n - 2)/(n + 1);.

1, 3, -3, 12, -24, 4, 60, -180, 65, -5, 360, -1440, 822, -132, 6, 2520, -12600, 9954, -2478, 231, -7, 20160, -120960, 122832, -41856, 5976, -368, 8, 181440, -1270080, 1581768, -688392, 133380, -12492, 549, -9, 1814400, -14515200, 21462480, -11412000, 2806200, -354720, 23610, -780, 10, 19958400

Offset

1,2

Comments

Table[Apply[Plus, CoefficientList[(n + 2)!P[x, n]/2, x]], {n, 0, 10}];

Row sums:

{1, 0, -8, -60, -384, -2380, -14208, -73836, -176000, 3824964, 104573760}

References

page 8 and page 42 - 43; Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986

Links

Table of n, a(n) for n=1..46.

Formula

a0=0; p(x,0)=1;p(x,1)=1+a0-x; P(x, n) = (2*n + a0 + 1 - x)*P(x, n - 1)/(n + 1) - n*P(x, n - 2)/(n + 1);

Example

{1},
{3, -3},
{12, -24, 4},
{60, -180,65, -5},
{360, -1440, 822, -132, 6},
{2520, -12600, 9954, -2478, 231, -7},
{20160, -120960,122832, -41856, 5976, -368, 8},
{181440, -1270080, 1581768, -688392, 133380, -12492, 549, -9},
{1814400, -14515200, 21462480, -11412000, 2806200, -354720, 23610, -780, 10}, {19958400, -179625600, 307937520, -193968720, 57998160, -9263760, 829290, -41382, 1067, -11},
{239500800, -2395008000, 4675026240, -3410138880, 1203543360, -232928640, 26271000, -1759104, 68388, -1416, 12}

Mathematica

a0 = 0; P[x, 0] = 1; P[x, 1] = 1 + a0 - x; P[x_, n_] := P[x, n] = (2*n + a0 + 1 - x)*P[x, n - 1]/(n + 1) - n*P[x, n - 2]/(n + 1); Table[ExpandAll[(n + 2)!*P[x, n]/2], {n, 0, 10}]; a = Table[CoefficientList[(n + 2)!*P[x, n]/2, x], {n, 0, 10}]; Flatten[a]

Crossrefs

Cf. A021009.

Sequence in context: A006804 A052533 A268798 * A268639 A192307 A328150

Adjacent sequences: A136530 A136531 A136532 * A136534 A136535 A136536

Keyword

uned,tabl,sign

Author

Roger L. Bagula, Mar 23 2008

Status

approved