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A008556
Triangle of coefficients of Legendre polynomials 2^n P_n (x).
3
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
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved