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Exponential convolution of central polygonal numbers (A000124) with themselves.
0

%I #30 May 08 2017 00:16:35

%S 1,4,16,62,230,812,2728,8752,26944,80000,230144,644096,1759744,

%T 4707328,12359680,31920128,81231872,204013568,506331136,1243217920,

%U 3022913536,7285243904,17415274496,41321234432,97370767360,227993976832,530713673728

%N Exponential convolution of central polygonal numbers (A000124) with themselves.

%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, 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]

%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (10,-40,80,-80,32)

%F O.g.f.: (1 - 6*x + 16*x^2 - 18*x^3 + 10*x^4)/(1 - 2*x)^5.

%F E.g.f.: (2 + 2*x + x^2)^2*exp(2*x)/4.

%F a(n) = 10*a(n-1) - 40*a(n-2) + 80*a(n-3) - 80*a(n-4) + 32*a(n-5).

%F a(n) = 2^(n - 6)*(n^4 + 2*n^3 + 19*n^2 + 42*n + 64).

%t LinearRecurrence[{10, -40, 80, -80, 32}, {1, 4, 16, 62, 230}, 27]

%t Table[2^(n - 6) (n^4 + 2 n^3 + 19 n^2 + 42 n + 64), {n, 0, 26}]

%Y Cf. A000124, A007465.

%K nonn,easy

%O 0,2

%A _Ilya Gutkovskiy_, Aug 27 2016