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A007558
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Shifts 2 places left when e.g.f. is squared.
(Formerly M1230)
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9
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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
(list;
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refs;
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history;
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OFFSET
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0,4
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REFERENCES
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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).
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LINKS
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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]
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FORMULA
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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
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MAPLE
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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:
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,nice,eigen
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AUTHOR
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STATUS
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approved
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