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A046716
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Coefficients of a special case of Poisson-Charlier polynomials.
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17
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1, 1, 1, 1, 3, 1, 1, 6, 8, 1, 1, 10, 29, 24, 1, 1, 15, 75, 145, 89, 1, 1, 21, 160, 545, 814, 415, 1, 1, 28, 301, 1575, 4179, 5243, 2372, 1, 1, 36, 518, 3836, 15659, 34860, 38618, 16072, 1, 1, 45, 834, 8274, 47775, 163191, 318926, 321690, 125673, 1, 1, 55, 1275, 16290, 125853, 606417, 1809905, 3197210, 2995011, 1112083, 1
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OFFSET
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0,5
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COMMENTS
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The sequence a(n) = Sum_{k = 0..n} T(n,k)*x^(n-k) is the binomial transform of the sequence b(n) = (n+x-1)! / (x-1)!. - Philippe Deléham, Jun 18 2004
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LINKS
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FORMULA
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Enneking and Ahuja reference gives the recurrence t(n, k) = t(n-1, k) - n*t(n-1, k-1) - (n-1)*t(n-2, k-2), with t(n, 0) = 1 and t(n, n) = (-1)^n. This sequence is T(n, k) = (-1)^k * t(n, k).
Let P(x, n) = Sum_{k = 0..n} T(n, k)*x^k, then Sum_{n>=0} P(x, n)*t^n / n! = exp(xt)/(1-xt)^(1/x). - Philippe Deléham, Jun 12 2004
T(n, 0) = 1, T(n, k) = (-1)^k * Sum_{i=n-k..n} (-1)^i*C(n, i)*S1(i, n-k), where S1 = Stirling numbers of first kind (A008275).
T(n, k) = T(n-1, k) + n*T(n-1, k-1) - (n-1)*T(n-2, k-2), with T(n, 0) = T(n, n) = 1.
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^(n+1)*A023443(n). (End)
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EXAMPLE
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Triangle starts:
1;
1, 1;
1, 3, 1;
1, 6, 8, 1;
1, 10, 29, 24, 1;
1, 15, 75, 145, 89, 1;
1, 21, 160, 545, 814, 415, 1;
1, 28, 301, 1575, 4179, 5243, 2372, 1;
1, 36, 518, 3836, 15659, 34860, 38618, 16072, 1;
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MAPLE
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a := proc(n, k) option remember;
if k = 0 then 1
elif k < 0 then 0
elif k = n then (-1)^n
else a(n-1, k) - n*a(n-1, k-1) - (n-1)*a(n-2, k-2) fi end:
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MATHEMATICA
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t[_, 0] = 1; t[n_, k_] := (-1)^k*Sum[(-1)^i*Binomial[n, i]*StirlingS1[i, n-k], {i, n-k, n}]; Table[t[n, k] // Abs, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 10 2014 *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, T[n-1, k] +n*T[n-1, k-1] - (n-1)*T[n-2, k-2]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 31 2024 *)
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PROG
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(Magma)
A046716:= func< n, k | (&+[(-1)^j*Binomial(n, k-j)*StirlingFirst(j+n-k, n-k): j in [0..k]]) >;
(SageMath)
def A046716(n, k): return sum(binomial(n, k-j)*stirling_number1(j+n-k, n-k) for j in range(k+1))
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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