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A137375
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Triangle read by rows, T(n,k) = (-1)^k*{{n,k}} where {{n,k}} are the second-order Stirling set numbers, n>=0, 0<=k<=n/2.
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6
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1, 0, 0, -1, 0, -1, 0, -1, 3, 0, -1, 10, 0, -1, 25, -15, 0, -1, 56, -105, 0, -1, 119, -490, 105, 0, -1, 246, -1918, 1260, 0, -1, 501, -6825, 9450, -945, 0, -1, 1012, -22935, 56980, -17325, 0, -1, 2035, -74316, 302995, -190575, 10395, 0, -1, 4082, -235092
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OFFSET
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0,9
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COMMENTS
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Also called "associated Stirling numbers of the second kind" by Riordan in the signless form with different offset A008299.
Old name was: Triangular sequence from coefficients of Mahler polynomials from expansion of: p(x) = exp(x*(1 + t - exp(t))) with weight n!:M(x,n).
M(n,x) = sum(k=0..n, (x)^k*sum(j=0..k, binomial(n,k-j)*stirling2(n-k+j,j)*(-1)^j)). - Vladimir Kruchinin, Jan 13 2012
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REFERENCES
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J. Riordan, Introduction to Combinatory Analysis, Wiley, New York, 1958.
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LINKS
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FORMULA
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T(n,k) = Sum_{j=0..k} C(n,k-j)*stirling2(n-k+j,j)*(-1)^(j). - Vladimir Kruchinin, Jan 13 2012
T(n,k) = (-1)^k*Sum_{j=0..n-k} C(j,n-2*k)*E2(n-k,n-k-j-1) for n>0, T(0,0) = 1, where E2(n,k) are the second-order Eulerian numbers A201637. - Peter Luschny, Nov 27 2012
Let p(x,t) = exp(x*(1+t-exp(t))) then T(n,k) = [x^k](n!*[t^n] series(p(x,t))) where [s^m] denotes the coefficient of s^m. - Peter Luschny, Dec 01 2012
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EXAMPLE
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[ 0] 1;
[ 1] 0;
[ 2] 0, -1;
[ 3] 0, -1;
[ 4] 0, -1, 3;
[ 5] 0, -1, 10;
[ 6] 0, -1, 25, -15;
[ 7] 0, -1, 56, -105;
[ 8] 0, -1, 119, -490, 105;
[ 9] 0, -1, 246, -1918, 1260;
[10] 0, -1, 501, -6825, 9450, -945;
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MAPLE
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A137375 := proc(n, k) if n = 0 then 1 else
add(binomial(j, n-2*k)* combinat[eulerian2](n-k, n-k-j-1), j=(0..n-k-1))*(-1)^k fi end: for n from 0 to 9 do seq(A137375(n, k), k=(0..n/2)) od; # Peter Luschny, Dec 01 2012
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MATHEMATICA
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Clear[p, x, t] p[t_] = Exp[x*(1 + t - Exp[t])]; Table[ ExpandAll[n!* SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}] a = Table[ CoefficientList[n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a];
Table[Sum[Binomial[n, k - j] StirlingS2[n - k + j, j] (-1)^j, {j, 0, k}], {n, 0, 15}, {k, 0, n/2}] // Flatten (* Eric W. Weisstein, Nov 13 2018 *)
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PROG
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(Maxima) T(n, k):=sum(binomial(n, k-j)*stirling2(n-k+j, j)*(-1)^(j), j, 0, k); /* Vladimir Kruchinin, Jan 13 2012 */
(Sage)
def A137375(n, k): return add(binomial(n, k-j)*(-1)^j*stirling_number2(n-k+j, j) for j in (0..k))
for n in range(11):
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CROSSREFS
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KEYWORD
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tabf,sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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