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

%I A008556 #26 Jul 08 2025 01:05:17
%S A008556 1,2,6,2,20,12,70,60,6,252,280,60,924,1260,420,20,3432,5544,2520,280,
%T A008556 12870,24024,13860,2520,70,48620,102960,72072,18480,1260,184756,
%U A008556 437580,360360,120120,13860,252,705432,1847560,1750320,720720,120120,5544
%N A008556 Triangle of coefficients of Legendre polynomials 2^n P_n (x).
%D A008556 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.
%H A008556 Robert Israel, <a href="/A008556/b008556.txt">Table of n, a(n) for n = 0..10099</a> (rows 0 to 199, flattened)
%H A008556 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F A008556 T(n,k) = C(2*(n-k), n-k) * C(n-k, k). - _Ralf Stephan_, Apr 07 2016
%e A008556 Triangle begins:
%e A008556 1,
%e A008556 2,
%e A008556 6, 2,
%e A008556 20, 12,
%e A008556 70, 60, 6,
%e A008556 252, 280, 60,
%e A008556 924, 1260, 420, 20,
%e A008556 3432, 5544, 2520, 280,
%e A008556 12870, 24024, 13860, 2520, 70,
%e A008556 48620, 102960, 72072, 18480, 1260,
%e A008556 184756, 437580, 360360, 120120, 13860, 252,
%e A008556 705432, 1847560, 1750320, 720720, 120120, 5544,
%e A008556 2704156, 7759752, 8314020, 4084080, 900900, 72072, 924,
%e A008556 10400600, 32449872, 38798760, 22170720, 6126120, 720720, 24024,
%e A008556 40116600, 135207800, 178474296, 116396280, 38798760, 6126120, 360360, 3432,
%e A008556 155117520, 561632400, 811246800, 594914320, 232792560, 46558512, 4084080, 102960,
%e A008556 ...
%p A008556 series(1/sqrt(1-2*x*z+z^2),z,20): for n to 19 do print(2^n*coeff(%,z,n)); od;
%t A008556 Table[Binomial[2 (n - k), n - k] Binomial[n - k, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten (* or *)
%t A008556 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 *)
%o A008556 (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
%o A008556 (PARI) T(n,k) = binomial(2*(n-k), n-k) * binomial(n-k, k);
%o A008556 for(n=0,10,for(k=0,n\2,print1(T(n,k),", "))); \\ _Joerg Arndt_, Apr 07 2016
%Y A008556 Cf. A115951.
%K A008556 nonn,tabf
%O A008556 0,2
%A A008556 _N. J. A. Sloane_