A259456 Triangle read by rows, giving coefficients in an expansion of absolute values of Stirling numbers of the first kind in terms of binomial coefficients.

1, 2, 3, 6, 20, 15, 24, 130, 210, 105, 120, 924, 2380, 2520, 945, 720, 7308, 26432, 44100, 34650, 10395, 5040, 64224, 303660, 705320, 866250, 540540, 135135, 40320, 623376, 3678840, 11098780, 18858840, 18288270, 9459450, 2027025, 362880, 6636960, 47324376, 177331440, 389449060, 520059540, 416215800

Offset: 0

Author

N. J. A. Sloane, Jun 30 2015

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins:
1,
2,3,
6,20,15,
24,130,210,105,
120,924,2380,2520,945,
...
For k=4 and j=2 in Knuth's equation, |S1(4,4-2)| = |S1(4,2)| = |A008275(4,2)| = 11 = p_{2,1}*C(4,3) +p_{2,2}*C(4,4) = 2*4+3*1. - _R. J. Mathar_, Jul 16 2015
		

References

• L. Comtet, Advanced Combinatorics (1974), Chapter VI, page 256.
• DJ Jeffrey, GA Kalugin, N Murdoch, Lagrange inversion and Lambert W, Preprint 2015; http://www.apmaths.uwo.ca/~djeffrey/Offprints/JeffreySYNASC2015paper17.pdf
• Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 152. Table C_{m, nu}.

Links

• Lothar Berg, On polynomials related with generalized Bernoulli numbers, Rostock Math. Kolloq. (2002).
• Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. 13 (2010) #10.4.4 page 4.
• Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
• Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 6.
• Donald E. Knuth, Convolution polynomials, The Mathematica J., 2 (1992), 67-78.
• Donald E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA], 1992.
• Richard B. Paris, An asymptotic approximation for incomplete Gaussian sums. II., J. Comp. Appl. Math 212 (2008) 16-30, Table 1.
• Grzegorz Rzadkowski, On some expansions for the Euler Gamma function and the Riemann Zeta function, arxiv:1007.1955 [math.CA], Table 1. J. Comp. Appl. Math. 236 (15) (2012), 3710-3719.
• Lajos Takács, On the number of distinct forests, SIAM J. Discrete Math., 3 (1990), 574-581. Table 3 gives a version of the triangle.

Crossrefs

Cf. This is a row reversed and unsigned version of A111999.

Cf. A008275, A000276 (2nd column), A000483 (3rd column), A000142 (1st column).

Cf. A133932.

Programs

Maple

A259456 := proc(n,k)
    option remember;
    if k < 1 or k > n  then
        0 ;
    elif n = 1 then
        1;
    else
        procname(n-1,k-1)+procname(n-1,k);
        %*(n+k-1) ;
    end if;
end proc:
seq(seq(A259456(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Jul 18 2015

Mathematica

T[n_, k_] := T[n, k] = If[k < 1 || k > n, 0, If[n == 1, 1, (T[n-1, k-1] + T[n-1, k])(n+k-1)]];
Table[T[n, k], {n, 1, 10}, { k, 1, n}] // Flatten (* Jean-François Alcover, Sep 26 2019, from Maple *)

Formula

T(n,k) = (n-k-1)*( T(n-1,k-1)+T(n-1,k) ), n>=1, 1<=k<=n. [Berg, Eq. 6]

The general results on the convolution of the refined partition polynomials of A133932, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these unsigned polynomials. - Tom Copeland, Sep 20 2016