OFFSET
0,2
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
L. Berg, On polynomials related with generalized Bernoulli numbers, Rostock Math. Kolloq. (2002).
S. Butler, P. Karasik, A note on nested sums, J. Int. Seq. 13 (2010) #10.4.4 page 4.
D. E. Knuth, Convolution polynomials, The Mathematica J., 2 (1992), 67-78.
D. E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA], 1992.
R. B. Paris, An asymptotic approximation for incomplete Gaussian sums. II., J. Comp. Appl. Math 212 (2008) 16-30, Table 1.
G. 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.
L. Takacs, On the number of distinct forests, SIAM J. Discrete Math., 3 (1990), 574-581. Table 3 gives a version of the triangle.
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
EXAMPLE
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
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 *)
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, Jun 30 2015
STATUS
approved