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

%I A154715 #33 May 12 2025 11:54:04
%S A154715 1,2,3,4,18,16,8,81,192,125,16,324,1536,2500,1296,32,1215,10240,31250,
%T A154715 38880,16807,64,4374,61440,312500,699840,705894,262144,128,15309,
%U A154715 344064,2734375,9797760,17294403,14680064,4782969
%N A154715 Triangle interpolating between the subsets of an n-set (A000079) and the trees on n labeled nodes (A000272) (read by rows).
%C A154715 Formatted as a square array:
%C A154715 1st row is A000079(n). Subsets of an n-set.
%C A154715 2nd row is A036290(n+1). Special (n+1)-subsets of a 3n-set partitioned into 3-blocks.
%C A154715 2nd column is A066274(n+1). Endofunctions of [n] such that 1 is not a fixed point.
%C A154715 1st column is A000272(n+2). Trees on n labeled nodes (Cayley's formula).
%C A154715 Alternating sum of rows in the triangle, Sum_{k=0..n} (-1)^(n-k) * T(n,k) = n! = A000142(n).
%C A154715 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 A154715 G. C. Greubel, <a href="/A154715/b154715.txt">Rows n = 0..100 of triangle, flattened</a>
%H A154715 Harlan J. Brothers, <a href="https://doi.org/10.35834/2025/3701067">Pascal's triangle, Sidi polynomials, and powers of e</a>, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.
%H A154715 Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>
%H A154715 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 A154715 T(n,k) = binomial(n,k)*(k+2)^n, where n >= 0, and k >= 0.
%F A154715 From _Wolfdieter Lang_, Oct 20 2022: (Start)
%F A154715 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 A154715 E.g.f. of column k: exp((k+2)*x)*((k+2)*x)^k/k!, for k >= 0. (End)
%F A154715 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 A154715 Triangle begins as:
%e A154715    1;
%e A154715    2,    3;
%e A154715    4,   18,    16;
%e A154715    8,   81,   192,    125;
%e A154715   16,  324,  1536,   2500,   1296;
%e A154715   32, 1215, 10240,  31250,  38880,  16807;
%e A154715   64, 4374, 61440, 312500, 699840, 705894, 262144;
%p A154715 T := proc(n,k) binomial(n,k)*(k+2)^n end;
%t A154715 Table[Binomial[n, k]*(k+2)^n, {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 09 2019 *)
%o A154715 (PARI) {T(n, k) = binomial(n,k)*(k+2)^n}; \\ _G. C. Greubel_, May 09 2019
%o A154715 (Magma) [[Binomial(n,k)*(k+2)^n: k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, May 09 2019
%o A154715 (Sage) [[binomial(n,k)*(k+2)^n for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, May 09 2019
%o A154715 (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*(k+2)^n ))); # _G. C. Greubel_, May 09 2019
%Y A154715 Cf. A000079, A000272, A036290, A066274, A075513.
%K A154715 easy,nonn,tabl
%O A154715 0,2
%A A154715 _Peter Luschny_, Jan 14 2009