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
Examples
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, ...
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].
Crossrefs
Cf. A115951.
Programs
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Maple
series(1/sqrt(1-2*x*z+z^2),z,20): for n to 19 do print(2^n*coeff(%,z,n)); od;
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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 *) -
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
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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
Formula
T(n,k) = C(2*(n-k), n-k) * C(n-k, k). - Ralf Stephan, Apr 07 2016