A154715 - OEIS

%I #23 Oct 30 2022 17:15:43

%S 1,2,3,4,18,16,8,81,192,125,16,324,1536,2500,1296,32,1215,10240,31250,

%T 38880,16807,64,4374,61440,312500,699840,705894,262144,128,15309,

%U 344064,2734375,9797760,17294403,14680064,4782969

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

%C Formatted as a square array:

%C 1st row is A000079(n). Subsets of an n-set.

%C 2nd row is A036290(n+1). Special (n+1)-subsets of a 3n-set partitioned into 3-blocks.

%C 2nd column is A066274(n+1). Endofunctions of [n] such that 1 is not a fixed point.

%C 1st column is A000272(n+2). Trees on n labeled nodes (Cayley's formula).

%C Alternating sum of rows in the triangle, Sum{k=0..n} (-1)^(n-k) * T(n,k)) = n! (A000142(n)).

%C 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

%H G. C. Greubel, <a href="/A154715/b154715.txt">Rows n = 0..100 of triangle, flattened</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>

%H Avram Sidi, <a href="https://doi.org/10.1090/S0025-5718-1980-0572861-2">Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities</a>, Math. Comp., 35 (1980), 851-874. Eq. (4.10), p. 862.

%F T(n,k) = binomial(n,k)*(k+2)^n, where n >= 0, and k >= 0.

%F From _Wolfdieter Lang_, Oct 20 2022: (Start)

%F 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.

%F E.g.f. of column k: exp((k+2)*x)*((k+2)*x)^k/k!, for k >= 0. (End)

%F 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

%e Triangle begins as:

%e    1;

%e    2,    3;

%e    4,   18,    16;

%e    8,   81,   192,    125;

%e   16,  324,  1536,   2500,   1296;

%e   32, 1215, 10240,  31250,  38880,  16807;

%e   64, 4374, 61440, 312500, 699840, 705894, 262144;

%p T := proc(n,k) binomial(n,k)*(k+2)^n end;

%t Table[Binomial[n, k]*(k+2)^n, {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 09 2019 *)

%o (PARI) {T(n, k) = binomial(n,k)*(k+2)^n}; \\ _G. C. Greubel_, May 09 2019

%o (Magma) [[Binomial(n,k)*(k+2)^n: k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, May 09 2019

%o (Sage) [[binomial(n,k)*(k+2)^n for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, May 09 2019

%o (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*(k+2)^n ))); # _G. C. Greubel_, May 09 2019

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

%K easy,nonn,tabl

%O 0,2

%A _Peter Luschny_, Jan 14 2009