A007558 Shifts 2 places left when e.g.f. is squared.

1, 1, 1, 2, 4, 10, 30, 100, 380, 1600, 7400, 37400, 204600, 1205600, 7612000, 51260000, 366784000, 2778820000, 22222332000, 187067320000, 1653461480000, 15310662400000, 148217381840000, 1497226615280000, 15754506226800000, 172407188412800000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• eigen

References

• O Bodini, M Dien, X Fontaine, A Genitrini, H K Hwang, Increasing Diamonds, in LATIN 2016: 12th Latin American Symposium, Ensenada, Mexico, April 11-15, 2016, Proceedings Pages pp 207-219 2016 DOI 10.1007/978-3-662-49529-2_16 Lecture Notes in Computer Science Series Volume 9644
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..518 (first 200 terms from Alois P. Heinz)
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Programs

Maple

a:= proc(n) option remember;
      `if`(n<2, 1, add(a(i)*a(n-2-i) *binomial(n-2, i), i=0..n-2))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 22 2012

Mathematica

a[n_] := a[n] = If[n < 2, 1, Sum[a[i] * a[n - 2 - i] * Binomial[n - 2, i], {i, 0, n - 2}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 03 2014, after Alois P. Heinz *)
Table[SeriesCoefficient[1 + (18 (WeierstrassP[x, {0, -1/108}] - WeierstrassPPrime[x, {0, -1/108}]))/(6 WeierstrassP[x, {0, -1/108}] - 1)^2, {x, 0, k}] k!, {k, 0, 30}] (* Jan Mangaldan, Nov 27 2020 *)

Formula

a(n) ~ c * d^n * n! * n, where d = 0.42089835222875301896706732846764190595145230471243866202153775712470703269... is the root of the equation WeierstrassP(1/d, 0, -1/108) = 1/6 and c = 1.06293253745327664869312823202016275205862332741406172188742740834633... - Vaclav Kotesovec, Sep 06 2014, updated Nov 27 2020

E.g.f.: 6^(1/3) * WeierstrassP((x+c)/6^(1/3), 0, -1/3), where c = 9.1898572290187191497581591181140131456801040793456712149069964791654... is the root of the equation WeierstrassP(c/6^(1/3), 0, -1/3) = 6^(-1/3). - Vaclav Kotesovec, Jun 14 2015

E.g.f. A(x) satisfies: A(x) = 1 + x + Integral(Integral A(x)^2 dx) dx. - Ilya Gutkovskiy, Jul 04 2020