A137375 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.

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

Offset: 0

Author

Roger L. Bagula, Apr 09 2008

Keywords

• tabf
• sign

Comments

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

Examples

			[ 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;
		

References

• J. Riordan, Introduction to Combinatory Analysis, Wiley, New York, 1958.

Links

• L. Carlitz, The coefficients in an asymptotic expansion, Proc. Amer. Math. Soc. 16 (1965) 248-252.
• A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.
• V. Kruchinin, D. Kruchinin, Application of a composition of generating functions for obtaining explicit formulas of polynomials, arXiv: 1211.0099 [math.NT], 2012.
• D. V. Kruchinin and V. V. Kruchinin, Explicit Formulas for Some Generalized Polynomials, Appl. Math. Inf. Sci. 7, No. 5, 2083-2088 (2013).
• Andrew Elvey Price, Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
• L. M. Smiley, Completion of a rational function sequence of Carlitz, arXiv:math/0006106 [math.CO], 2000.
• E. W. Weisstein, Mahler Polynomial.

Crossrefs

Row sums are: A000587.

Programs

Maple

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

Mathematica

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 *)

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):
    [A137375(n, k) for k in (0..n//2)]  # Peter Luschny, Dec 01 2012

Formula

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

Extensions

Edited and simpler definition by Peter Luschny, Nov 27 2012