A005492 From expansion of falling factorials.

4, 15, 52, 151, 372, 799, 1540, 2727, 4516, 7087, 10644, 15415, 21652, 29631, 39652, 52039, 67140, 85327, 106996, 132567, 162484, 197215, 237252, 283111, 335332, 394479, 461140, 535927, 619476, 712447, 815524, 929415, 1054852, 1192591, 1343412, 1508119, 1687540

Offset: 4

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 4..1004
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• E. G. Whitehead, Jr., Stirling number identities from chromatic polynomials, J. Combin. Theory, A 24 (1978), 314-317.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Row n=4 of A108087 (shifted and first term prepended).

Cf. A005490.

Programs

Magma

[n^4 -16*n^3 +102*n^2 -300*n +340: n in [4..50]]; // G. C. Greubel, Dec 01 2022

Maple

A005492:=-(15-23*z+41*z**2-13*z**3+4*z**4)/(z-1)**5; # Conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence except for the leading 4.

Mathematica

LinearRecurrence[{5,-10,10,-5,1},{4,15,52,151,372},50] (* Harvey P. Dale, Dec 25 2012 *)

SageMath

[n^4 -16*n^3 +102*n^2 -300*n +340 for n in range(4,51)] # G. C. Greubel, Dec 01 2022

Formula

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

a(n) = n^4 - 16*n^3 + 102*n^2 - 300*n + 340.

G.f.: x^4*(4-5*x+17*x^2+x^3+7*x^4)/(1-x)^5. - Harvey P. Dale, Dec 25 2012

E.g.f.: (1/6)*(-2040 - 762*x - 108*x^2 - 7*x^3 + (2040 - 1278*x + 366*x^2 - 60*x^3 + 6*x^4)*exp(x)). - G. C. Greubel, Dec 01 2022

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004