A154715 Triangle interpolating between the subsets of an n-set (A000079) and the trees on n labeled nodes (A000272) (read by rows).

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, 128, 15309, 344064, 2734375, 9797760, 17294403, 14680064, 4782969

Offset: 0

Author

Peter Luschny, Jan 14 2009

Keywords

• easy
• nonn
• tabl

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

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.
• 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.

Crossrefs

Cf. A000079, A000272, A036290, A066274, A075513.

Programs

GAP

Flat(List([0..12], n-> List([0..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

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 *)

PARI

{T(n, k) = binomial(n,k)*(k+2)^n}; \\ 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

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