A112487 a(n) = Sum_{k=0..n} E2(n, k)*2^k, where E2(n, k) are the second-order Eulerian numbers A340556.
1, 2, 10, 82, 938, 13778, 247210, 5240338, 128149802, 3551246162, 109979486890, 3764281873042, 141104799067178, 5749087305575378, 252969604725106090, 11955367835505775378, 603967991604199335722, 32479636694930586142802, 1852497140997527094395050
Offset: 0
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..376
- Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
- Kenny Barrese, Jennifer Elder, Pamela E. Harris, and Anthony Simpson, Enumerating Flat Fubini Rankings, arXiv:2504.06466 [math.CO], 2025. Conjecture 4.4. p. 14.
- W. Steven Gray, Luis A. Duffaut Espinosa, and Kurusch Ebrahimi-Fard, Additive Networks of Chen-Fliess Series: Local Convergence and Relative Degree, arXiv:2104.08950 [eess.SY], 2021.
- W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
- M. Thitsa and W. S. Gray, On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems, SIAM Journal on Control and Optimization, Vol. 50, No. 5, pp. 2786-2813. - From _N. J. A. Sloane_, Dec 26 2012
Programs
-
Maple
A112487 := proc(n) add(A112486(n,k),k=0..n) ; end proc: # R. J. Mathar, Dec 19 2013 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 E2 := (n, k) -> `if`(k=0, k^n, combinat:-eulerian2(n, k-1)); a := n -> add(E2(n, k)*2^k, k=0..n): seq(a(n), n=0..17); # Peter Luschny, Feb 13 2021
-
Mathematica
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[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}]; Table[a[n], {n, 0, 17}] (* Peter Luschny, Feb 13 2021 *) -
Maxima
a(n):=n!*(sum(binomial(n+k, n)*sum((-1)^(n+j)*binomial(k, j)*sum((binomial(j, l)*(j-l)!*2^(j-l)*(-1)^l*stirling2(n-l+j, j-l))/(n-l+j)!, l, 0, j), j, 0, k), k, 0, n)); /* Vladimir Kruchinin, Feb 14 2012 */
-
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
Formula
a(n) = Sum_{m=0..n} A112486(n, m), n >= 0.
a(n) = 2*A032188(n+1), n > 0. - Vladeta Jovovic, Jul 11 2007
From Paul D. Hanna, Jun 30 2009: (Start)
E.g.f. A(x) satisfies: A'(x) = A(x)^2 + A(x)^3.
E.g.f. A(x) satisfies: A(x) = exp( Integral[A(x) + A(x)^2]dx ) with A(0)=1. (End)
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
From Tom Copeland, Oct 05 2011: (Start)
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).
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.
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)).
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)
a(n) = n!*Sum_{k=0..n} binomial(n+k, n)*Sum_{j=0..k} (-1)^(n+j)*binomial(k, j)*Sum_{l=0..j} binomial(j, l)*(j-l)!*2^(j-l)*(-1)^l*Stirling2(n-l+j, j-l)/(n-l+j)!. - Vladimir Kruchinin, Feb 14 2012
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
a(n) ~ n^n / (exp(n) * (1-log(2))^(n+1/2)). - Vaclav Kotesovec, Aug 14 2017
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
Extensions
New name from Peter Luschny, Feb 13 2021
Comments