cp's OEIS Frontend

A294221 Exponential transform of the square pyramidal numbers (A000330).

%I A294221 #5 Feb 16 2025 08:33:51
%S A294221 1,1,6,30,192,1471,12637,120723,1267492,14438913,176961001,2318180239,
%T A294221 32275104644,475285152707,7373223596299,120078748361611,
%U A294221 2046720320727328,36414341169682417,674650306604656821,12988470845576660407,259348785562811740236,5361803880323803698731,114593610390850499426211
%N A294221 Exponential transform of the square pyramidal numbers (A000330).
%H A294221 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 A294221 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 A294221 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A294221 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ExponentialTransform.html">Exponential Transform</a>
%H A294221 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquarePyramidalNumber.html">Square Pyramidal Number</a>
%H A294221 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%F A294221 E.g.f.: exp(exp(x)*x*(6 + 9*x + 2*x^2)/6).
%e A294221 E.g.f.: A(x) = 1 + x/1! + 6*x^2/2! + 30*x^3/3! + 192*x^4/4! + 1471*x^5/5! + 12637*x^6/6! + ...
%t A294221 Range[0, 22]! CoefficientList[Series[Exp[Exp[x] x (6 + 9 x + 2 x^2)/6], {x, 0, 22}], x]
%t A294221 a[n_] := a[n] = Sum[a[n - k] Binomial[n - 1, k - 1] k (k + 1) (2 k + 1)/6, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]
%Y A294221 Cf. A000330, A033462, A279361, A281231.
%K A294221 nonn
%O A294221 0,3
%A A294221 _Ilya Gutkovskiy_, Oct 25 2017