A294221 Exponential transform of the square pyramidal numbers (A000330).
1, 1, 6, 30, 192, 1471, 12637, 120723, 1267492, 14438913, 176961001, 2318180239, 32275104644, 475285152707, 7373223596299, 120078748361611, 2046720320727328, 36414341169682417, 674650306604656821, 12988470845576660407, 259348785562811740236, 5361803880323803698731, 114593610390850499426211
Offset: 0
Keywords
Examples
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! + ...
Links
- 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 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]
- N. J. A. Sloane, Transforms
- Eric Weisstein's World of Mathematics, Exponential Transform
- Eric Weisstein's World of Mathematics, Square Pyramidal Number
- Index to sequences related to pyramidal numbers
Programs
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Mathematica
Range[0, 22]! CoefficientList[Series[Exp[Exp[x] x (6 + 9 x + 2 x^2)/6], {x, 0, 22}], x] 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}]
Formula
E.g.f.: exp(exp(x)*x*(6 + 9*x + 2*x^2)/6).