A179495 E.g.f. satisfies: A'(x) = [A(x)^2 + A(x)^3]/(x^2 + x^3).

0, 1, 2, 12, 84, 820, 9540, 132888, 2129232, 38760048, 788500800, 17740459440, 437238410400, 11716457100192, 339129808346784, 10544636706428160, 350515939418507520, 12404398847785793280, 465618362609300313600

Offset

0,3

Comments

a(n) = n * A179496(n-1). - Vaclav Kotesovec, Dec 25 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From N. J. A. Sloane, Feb 06 2013

Formula

E.g.f. satisfies: d/dx A_n(x) = [A_n(x)^2 + A_n(x)^3]/(x^2 + x^3) where A_n(x) denotes the n-th iteration of e.g.f. A(x).

...

Define a triangular matrix where the e.g.f. of column k equals (A(x)/x)^k, then the matrix log is the matrix L with L(n+1,n)=L(n+2,n)=n+1 and zeros elsewhere.

a(n) ~ sqrt(1+r) * n^n * r^(n-1) / exp(n), where r = -1-LambertW(-1, -exp(-2)) = 2.146193220620582585237... is the root of the equation log(1+r)=r-1. - Vaclav Kotesovec, Jan 04 2014

Example

E.g.f. A(x) = x + 2*x^2/2! + 12*x^3/3! + 84*x^4/4! + 820*x^5/5! +...
Related expansions:
A(x)^2 + A(x)^3 = 2*x^2/2! + 18*x^3/3! + 192*x^4/4! + 2400*x^5/5! +...
A'(x) = 1 + 2*x + 12*x^2/2! + 84*x^3/3! + 820*x^4/4! + 9540*x^5/5! +...
A(x)/x = 1 + x + 4*x^2/2! + 21*x^3/3! + 164*x^4/4! + 1590*x^5/5! +...
...
Define a triangular matrix where the e.g.f. of column k equals A(x)^k:
1;
1, 1;
4/2!, 2, 1;
21/3!, 10/2!, 3, 1;
164/4!, 66/3!, 18/2!, 4, 1;
1590/5!, 592/4!, 141/3!, 28/2!, 5, 1;
18984/6!, 6500/5!, 1428/4!, 252/3!, 40/2!, 6, 1;
266154/7!, 85548/6!, 17430/5!, 2840/4!, 405/3!, 54/2!, 7, 1;
...
then the matrix log of the above matrix equals:
0;
1, 0;
1, 2, 0;
0, 2, 3, 0;
0, 0, 3, 4, 0;
0, 0, 0, 4, 5, 0;
0, 0, 0, 0, 5, 6, 0; ...

Mathematica

nmax = 20; aa = ConstantArray[0, nmax]; aa[[1]] = 1; aa[[2]] = 2; Do[AGF = Sum[aa[[n]]*x^n/n!, {n, 1, j - 1}] + koef*x^j/j!; sol = Solve[Coefficient[D[AGF, x]*(x^2 + x^3) - (AGF^2 + AGF^3), x, j + 1] == 0, koef][[1]]; aa[[j]] = koef /. sol[[1]], {j, 3, nmax}]; Flatten[{0, aa}] (* Vaclav Kotesovec, Dec 25 2013 *)

Prog

(PARI) {a(n)=local(A=x+x^2+O(x^(n+1)), D=1); n!*polcoeff(1+sum(m=1, n+1, (D=A*deriv(x*D+O(x^(n+1))))/m!), n-1)}

Crossrefs

Cf. A179496.

Sequence in context: A052887 A052867 A226238 * A348765 A364279 A208977

Adjacent sequences: A179492 A179493 A179494 * A179496 A179497 A179498

Keyword

nonn

Author

Paul D. Hanna, Jul 25 2010

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014

Status

approved