A008556 Triangle of coefficients of Legendre polynomials 2^n P_n (x).

1, 2, 6, 2, 20, 12, 70, 60, 6, 252, 280, 60, 924, 1260, 420, 20, 3432, 5544, 2520, 280, 12870, 24024, 13860, 2520, 70, 48620, 102960, 72072, 18480, 1260, 184756, 437580, 360360, 120120, 13860, 252, 705432, 1847560, 1750320, 720720, 120120, 5544

Offset

0,2

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.

Links

Robert Israel, Table of n, a(n) for n = 0..10099 (rows 0 to 199, flattened)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Formula

T(n,k) = C(2*(n-k), n-k) * C(n-k, k). - Ralf Stephan, Apr 07 2016

Example

Triangle begins:
1,
2,
6, 2,
20, 12,
70, 60, 6,
252, 280, 60,
924, 1260, 420, 20,
3432, 5544, 2520, 280,
12870, 24024, 13860, 2520, 70,
48620, 102960, 72072, 18480, 1260,
184756, 437580, 360360, 120120, 13860, 252,
705432, 1847560, 1750320, 720720, 120120, 5544,
2704156, 7759752, 8314020, 4084080, 900900, 72072, 924,
10400600, 32449872, 38798760, 22170720, 6126120, 720720, 24024,
40116600, 135207800, 178474296, 116396280, 38798760, 6126120, 360360, 3432,
155117520, 561632400, 811246800, 594914320, 232792560, 46558512, 4084080, 102960,
...

Maple

series(1/sqrt(1-2*x*z+z^2), z, 20): for n to 19 do print(2^n*coeff(%, z, n)); od;

Mathematica

Table[Binomial[2 (n - k), n - k] Binomial[n - k, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten (* or *)
Table[Reverse@ Abs@ CoefficientList[Series[2^n LegendreP[n, x], {x, 0, n}], x] /. 0 -> Nothing, {n, 0, 11}] // Flatten (* Michael De Vlieger, Apr 07 2016 *)

Prog

(PARI) row(n) = my(v = Vec(2^n*pollegendre(n))); vector((#v+1)\2, k, abs(v[2*k-1])); \\ Michel Marcus, Apr 07 2016
(PARI) T(n, k) = binomial(2*(n-k), n-k) * binomial(n-k, k);
for(n=0, 10, for(k=0, n\2, print1(T(n, k), ", "))); \\ Joerg Arndt, Apr 07 2016

Crossrefs

Cf. A115951.

Sequence in context: A248568 A257252 A327171 * A254638 A320118 A320016

Adjacent sequences: A008553 A008554 A008555 * A008557 A008558 A008559

Keyword

nonn,tabf

Author

N. J. A. Sloane.

Status

approved