A007558 - OEIS

%I M1230 #50 Oct 30 2023 00:13:27

%S 1,1,1,2,4,10,30,100,380,1600,7400,37400,204600,1205600,7612000,

%T 51260000,366784000,2778820000,22222332000,187067320000,1653461480000,

%U 15310662400000,148217381840000,1497226615280000,15754506226800000,172407188412800000

%N Shifts 2 places left when e.g.f. is squared.

%D 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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vaclav Kotesovec, <a href="/A007558/b007558.txt">Table of n, a(n) for n = 0..518</a> (first 200 terms from Alois P. Heinz)

%H M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

%F 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

%F 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

%F E.g.f. A(x) satisfies: A(x) = 1 + x + Integral(Integral A(x)^2 dx) dx. - _Ilya Gutkovskiy_, Jul 04 2020

%p a:= proc(n) option remember;

%p       `if`(n<2, 1, add(a(i)*a(n-2-i) *binomial(n-2, i), i=0..n-2))

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Jun 22 2012

%t 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_ *)

%t 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 *)

%K nonn,nice,eigen

%O 0,4

%A _N. J. A. Sloane_