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Search: id:A137477
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| A137477 |
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A triangular sequence of coefficients from the inverse substitution of in the spherical Bessel polynomial recursion:k=1;x->1/y; B(x, n) = (-2/x)*B(x, n - 1) - (k^2 - (n*(n - 1)/x^2))*B(x, n - 2). |
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1
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| 1, 0, -2, -1, 0, 6, 0, 4, 0, -24, 1, 0, -26, 0, 120, 0, -6, 0, 156, 0, -720, -1, 0, 68, 0, -1212, 0, 5040, 0, 8, 0, -544, 0, 9696, 0, -40320, 1, 0, -140, 0, 6108, 0, -92304, 0, 362880, 0, -10, 0, 1400, 0, -61080, 0, 923040, 0, -3628800, -1, 0, 250, 0, -21508, 0, 764184, 0, -10516320, 0, 39916800
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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Row sums are;
{1, -2, 5, -20, 95, -570, 3895, -31160, 276545, -2765450, 30143405}
These polynomials are closely related to Yakawa wave functions
in meson nuclear force theory.
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REFERENCES
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Weisstein, Eric W., Spherical Bessel Differential Equation, http://mathworld.wolfram.com/SphericalBesselDifferentialEquation.html
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FORMULA
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k=1;x->1/y; B(x, n) = (-2/x)*B(x, n - 1) - (k^2 - (n*(n - 1)/x^2))*B(x, n - 2).
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EXAMPLE
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{1},
{0, -2},
{-1, 0, 6},
{0, 4, 0, -24},
{1, 0, -26, 0, 120},
{0, -6,0, 156, 0, -720},
{-1, 0,68, 0, -1212, 0, 5040},
{0, 8, 0, -544, 0, 9696, 0, -40320},
{1, 0, -140, 0, 6108, 0, -92304, 0, 362880},
{0, -10, 0, 1400, 0, -61080, 0, 923040, 0, -3628800},
{-1, 0, 250, 0, -21508, 0, 764184, 0, -10516320, 0, 39916800}
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MATHEMATICA
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Clear[B, x] k = 1; B[x, -1] = 0; B[x, 0] = 1; B[x_, n_] := B[x, n] = (-2/x)*B[x, n - 1] - (k^2 - (n*(n - 1)/x^2))*B[x, n - 2]; g = Table[ExpandAll[B[x, n]], {n, 0, 10}]; a = Table[CoefficientList[B[x, n] /. x -> 1/y, y], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[B[x, n] /. x -> 1/y, y]], {n, 0, 10}];
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CROSSREFS
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Sequence in context: A130191 A059720 A140589 this_sequence A119275 A129462 A122930
Adjacent sequences: A137474 A137475 A137476 this_sequence A137478 A137479 A137480
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KEYWORD
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nonn,uned,tabl
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 21 2008
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