OFFSET
0,2
COMMENTS
Formatted as a square array:
1st row is A000079(n). Subsets of an n-set.
2nd row is A036290(n+1). Special (n+1)-subsets of a 3n-set partitioned into 3-blocks.
2nd column is A066274(n+1). Endofunctions of [n] such that 1 is not a fixed point.
1st column is A000272(n+2). Trees on n labeled nodes (Cayley's formula).
Alternating sum of rows in the triangle, Sum{k=0..n} (-1)^(n-k) * T(n,k)) = n! (A000142(n)).
This triangle gives the coefficient of Sidi's polynomials D_{n,2,n}(-z)/(-z), for n >= 0. See [Sidi 1980]. - Wolfdieter Lang, Oct 27 2022
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874. Eq. (4.10), p. 862.
FORMULA
T(n,k) = binomial(n,k)*(k+2)^n, where n >= 0, and k >= 0.
From Wolfdieter Lang, Oct 20 2022: (Start)
O.g.f. of column k: (-x)^k*(k + 2)^k/(1 - (k + 2)*x)^(k+1), for k >= 0. See |A075513| with offset 0.
E.g.f. of column k: exp((k+2)*x)*((k+2)*x)^k/k!, for k >= 0. (End)
E.g.f. of triangle (of row polynomials in y): exp(2*x)*substitute(z = x*y*exp(x), LambertW(-z)^2/(-z)*2*(1 + LambertW(-z)))). - Wolfdieter Lang, Oct 24 2022
EXAMPLE
Triangle begins as:
1;
2, 3;
4, 18, 16;
8, 81, 192, 125;
16, 324, 1536, 2500, 1296;
32, 1215, 10240, 31250, 38880, 16807;
64, 4374, 61440, 312500, 699840, 705894, 262144;
MAPLE
T := proc(n, k) binomial(n, k)*(k+2)^n end;
MATHEMATICA
Table[Binomial[n, k]*(k+2)^n, {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 09 2019 *)
PROG
(PARI) {T(n, k) = binomial(n, k)*(k+2)^n}; \\ G. C. Greubel, May 09 2019
(Magma) [[Binomial(n, k)*(k+2)^n: k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 09 2019
(Sage) [[binomial(n, k)*(k+2)^n for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 09 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n, k)*(k+2)^n ))); # G. C. Greubel, May 09 2019
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, Jan 14 2009
STATUS
approved