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A008556
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Triangle of coefficients of Legendre polynomials 2^n P_n (x).
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3
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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
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OFFSET
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0,2
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REFERENCES
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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.
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LINKS
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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].
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FORMULA
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T(n,k) = C(2*(n-k), n-k) * C(n-k, k). - Ralf Stephan, Apr 07 2016
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EXAMPLE
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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,
...
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MAPLE
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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
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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