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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 (list; graph; refs; listen; history; text; internal format)
 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 STATUS approved

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Last modified August 13 05:18 EDT 2022. Contains 356077 sequences. (Running on oeis4.)