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Exponential-convolution of triangular numbers with themselves.
(Formerly M4195)
1

%I M4195 #25 Feb 24 2019 09:08:19

%S 1,6,30,128,486,1692,5512,17040,50496,144512,401664,1089024,2890240,

%T 7529472,19298304,48754688,121602048,299827200,731643904,1768685568,

%U 4239261696,10081796096,23805296640,55839817728,130187001856,301813727232,696036360192,1597358735360

%N Exponential-convolution of triangular numbers with themselves.

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

%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 G.f.: (-1-6*x^4+12*x^3-10*x^2+4*x)/(2*x-1)^5. [Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009]

%F E.g.f.: (1/4)*exp(2*x)*(2 + 4*x + x^2)^2. - _Ilya Gutkovskiy_, Mar 21 2018

%t a = DifferenceRoot[Function[{a, n}, {(-2n^4 - 28n^3 - 158n^2 - 388n - 384)* a[n] + (n^4 + 10n^3 + 43n^2 + 74n + 64)*a[n+1] == 0, a[0] == 1}]];

%t Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Feb 24 2019 *)

%Y Cf. A000217.

%K nonn

%O 0,2

%A _N. J. A. Sloane_.