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A158706
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Expansion of e.g.f.: exp(t*x)/(1 - x/t - t^2 * x^2).
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3
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1, 1, 0, 1, 2, 0, 2, 0, 3, 6, 0, 6, 0, 15, 0, 7, 24, 0, 24, 0, 84, 0, 52, 0, 37, 120, 0, 120, 0, 540, 0, 380, 0, 485, 0, 141, 720, 0, 720, 0, 3960, 0, 3000, 0, 5430, 0, 2406, 0, 1111, 5040, 0, 5040, 0, 32760, 0, 26040, 0, 60690, 0, 32802, 0, 28147, 0, 5923
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OFFSET
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0,5
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LINKS
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FORMULA
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T(n, k) = coefficients of the expansion : p(x,t) = exp(t*x)/(1 - x/t - t^2* x^2).
T(n, k) = coefficients of the series : Sum_{s=0..n} Sum_{j=0..floor(s/2)} (n!/(n-s)!)*A011973(s, j)*x^(4*j+2*n-2*s). - G. C. Greubel, Nov 30 2021
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EXAMPLE
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Irregular triangle begins as:
1;
1, 0, 1;
2, 0, 2, 0, 3;
6, 0, 6, 0, 15, 0, 7;
24, 0, 24, 0, 84, 0, 52, 0, 37;
120, 0, 120, 0, 540, 0, 380, 0, 485, 0, 141;
720, 0, 720, 0, 3960, 0, 3000, 0, 5430, 0, 2406, 0, 1111;
5040, 0, 5040, 0, 32760, 0, 26040, 0, 60690, 0, 32802, 0, 28147, 0, 5923;
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MATHEMATICA
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(* First program *)
Table[CoefficientList[n!*t^n*SeriesCoefficient[Series[Exp[t*x]/(1 -x/t -t^2*x^2), {x, 0, 20}], n], t], {n, 0, 10}]//Flatten
(* Second program *)
Table[CoefficientList[Series[Sum[Sum[GegenbauerC[k, (s+1)/2 -k, 1]*x^(4*k+2*n - 2*s)*(n!/(n-s)!), {k, 0, Floor[s/2]}], {s, 0, n}], {x, 0, 20}], x], {n, 0, 10}] (* G. C. Greubel, Nov 30 2021 *)
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PROG
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(Sage)
@CachedFunction
def A011973(n, k): return 0 if (k<0 or k>(n//2)) else binomial(n-k, k)
def f(n, x): return sum( sum( (A011973(s, j)/factorial(n-s))*x^(4*j+2*n-2*s) for j in (0..(s//2)) ) for s in (0..n) )
def A158706(n, k): return factorial(n)*( f(n, x) ).series(x, 2*n+1).list()[k]
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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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EXTENSIONS
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STATUS
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approved
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