|
%I #71 Jul 23 2021 02:17:58
%S 1,2,10,82,938,13778,247210,5240338,128149802,3551246162,109979486890,
%T 3764281873042,141104799067178,5749087305575378,252969604725106090,
%U 11955367835505775378,603967991604199335722,32479636694930586142802,1852497140997527094395050
%N a(n) = Sum_{k=0..n} E2(n, k)*2^k, where E2(n, k) are the second-order Eulerian numbers A340556.
%C Previous name: Row sums of triangle A112486.
%H Michael De Vlieger, <a href="/A112487/b112487.txt">Table of n, a(n) for n = 0..376</a>
%H Roland Bacher, <a href="https://doi.org/10.37236/2522">Counting Packings of Generic Subsets in Finite Groups</a>, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
%H W. Steven Gray, Luis A. Duffaut Espinosa, and Kurusch Ebrahimi-Fard, <a href="https://arxiv.org/abs/2104.08950">Additive Networks of Chen-Fliess Series: Local Convergence and Relative Degree</a>, arXiv:2104.08950 [eess.SY], 2021.
%H W. S. Gray and M. Thitsa, <a href="https://doi.org/10.1109/SSST.2013.6524939">System Interconnections and Combinatorial Integer Sequences</a>, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
%H M. Thitsa and W. S. Gray, <a href="http://dx.doi.org/10.1137/110852760">On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems</a>, SIAM Journal on Control and Optimization, Vol. 50, No. 5, pp. 2786-2813. - From _N. J. A. Sloane_, Dec 26 2012
%F a(n) = Sum_{m=0..n} A112486(n, m), n >= 0.
%F a(n) = 2*A032188(n+1), n > 0. - _Vladeta Jovovic_, Jul 11 2007
%F From _Paul D. Hanna_, Jun 30 2009: (Start)
%F E.g.f. A(x) satisfies: A'(x) = A(x)^2 + A(x)^3.
%F E.g.f. A(x) satisfies: A(x) = exp( Integral[A(x) + A(x)^2]dx ) with A(0)=1. (End)
%F E.g.f. A(x) satisfies: A(x) = 2*exp(A(x)) - (2+x), where A(x) = Sum_{n>=0} a(n)*x^(n+1)/(n+1)! (the e.g.f. when offset=1). - _Paul D. Hanna_, Sep 23 2011
%F From _Tom Copeland_, Oct 05 2011: (Start)
%F With c(0)= 0 and c(n+1)= (-1)^n a(n) for n>=0, c(n)=(-1)^(n+1) PW(n,-2) with PW the Ward polynomials A134991. E.g.f. for the c(n) is A(x) = -(x+2)-LW{-2 exp[-(x+2)]}, where LW(x) is a suitable branch of the Lambert W Fct. (see A135338).
%F The compositional inverse is B(x) = x + 2(exp(x) - x - 1). These results are a special case of A134685 with u(x)=B(x), i.e., u_1=1 and (u_n)=2 for n>0.
%F Let h(x) = 1/(dB(x)/dx) = 1/[1+2(exp(x)-1)], then c(n) is given by (h(x)*d/dx)^n x, evaluated at x=0, i.e., A(x) = exp(x*h(u)*d/du) u, evaluated at u=0. Also, dA(x)/dx = h(A(x)).
%F The e.g.f. A(x) = -v * Sum_(j>=1) D(j-1,u) (-z)^j/ j! where u=-(x+2), v=1+u, z=(1+v)/(v^2) and D(j-1,u) are the polynomials of A042977. (End)
%F a(n) = (n-1)!*(Sum_{k=0..n-1} binomial(n+k-1, n-1)*Sum_{j=0..k} (-1)^(n+j-1)* binomial(k, j)*Sum_{l=0..j} binomial(j, l)*(j-l)!*2^(j-l)*(-1)^l*Stirling2(n-l+j-1, j-l))/(n-l+j-1)!)))), n>0. - _Vladimir Kruchinin_, Feb 14 2012
%F G.f.: 1/Q(0), where Q(k)= 1 + k*x - 2*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 01 2013
%F a(n) ~ n^n / (exp(n) * (1-log(2))^(n+1/2)). - _Vaclav Kotesovec_, Aug 14 2017
%F a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1). - _Ilya Gutkovskiy_, Jul 02 2020
%p A112487 := proc(n)
%p add(A112486(n,k),k=0..n) ;
%p end proc: # _R. J. Mathar_, Dec 19 2013
%p seq(op(k, convert(asympt(GAMMA(n, 2*n)*exp(2*n)/(2*n)^n, n, 20), polynom))*(-1)^(k+1)*n^k, k = 1..19); # Maple 2017, _Vaclav Kotesovec_, Aug 14 2017
%p E2 := (n, k) -> `if`(k=0, k^n, combinat:-eulerian2(n, k-1));
%p a := n -> add(E2(n, k)*2^k, k=0..n):
%p seq(a(n), n=0..17); # _Peter Luschny_, Feb 13 2021
%t a[n_] := (n-1)!*(Sum[ Binomial[n+k-1, n-1]* Sum[(-1)^(n+j-1)*Binomial[k, j]* Sum[(Binomial[j, l]*(j-l)!*2^(j-l)*(-1)^l*StirlingS2[n-l+j-1, j-l])/(n-l+j-1)!, {l, 0, j}], {j, 0, k}], {k, 0, n-1}]); Table[a[n], {n, 1, 18}] (* _Jean-François Alcover_, Feb 26 2013, after _Vladimir Kruchinin_ *)
%t T[n_, k_] := T[n, k] = If[k == 0, Boole[n == 0], If[n < 0, 0, k T[n - 1, k] + (2 n - k) T[n - 1, k - 1]]]; a[n_] := Sum[T[n, k] 2^k, {k, 0, n}];
%t Table[a[n], {n, 0, 17}] (* _Peter Luschny_, Feb 13 2021 *)
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(intformal(A+A^2)+x*O(x^n)));n!*polcoeff(A,n)} \\ _Paul D. Hanna_, Jun 30 2009
%o (Maxima) a(n):=(n-1)!*(sum(binomial(n+k-1, n-1)*sum((-1)^(n+j-1)*binomial(k, j)*sum((binomial(j, l)*(j-l)!*2^(j-l)*(-1)^l*stirling2(n-l+j-1, j-l))/(n-l+j-1)!, l, 0, j), j, 0, k), k, 0, n-1)); /* _Vladimir Kruchinin_, Feb 14 2012 */
%Y Cf. A340556, A112486, A032188, A032034, A134991, A135338.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 12 2005
%E New name from _Peter Luschny_, Feb 13 2021
|
