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

0,2

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

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

Sequence in context: A370868 A329545 A187075 * A077407 A273002 A333547

Adjacent sequences: A154712 A154713 A154714 * A154716 A154717 A154718

Keyword

easy,nonn,tabl

Author

Peter Luschny, Jan 14 2009

Status

approved