A002451 Expansion of 1/((1-x)*(1-4*x)*(1-9*x)).

1, 14, 147, 1408, 13013, 118482, 1071799, 9668036, 87099705, 784246870, 7059619931, 63542171784, 571901915677, 5147206719578, 46325218390143, 416928397167052, 3752361301126529, 33771274616631006, 303941563175648035, 2735474435084708240, 24619271381777877861

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 35.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)
• 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
• Index entries for linear recurrences with constant coefficients, signature (14,-49,36).

Crossrefs

A diagonal of A036969.

Programs

GAP

List([0..30],n->1/24-4^(n+2)/15+9^(n+2)/40); # Muniru A Asiru, Dec 18 2018

Magma

[(10 - 4^(n+4) +6*9^(n+2))/240: n in [0..30]]; // G. C. Greubel, Jul 04 2019

Maple

a:=n->sum((9^(n-j)-4^(n-j))/5,j=0..n): seq(a(n), n=1..30); # Zerinvary Lajos, Jan 15 2007
A002451:=-1/(z-1)/(4*z-1)/(9*z-1); # Simon Plouffe in his 1992 dissertation

Mathematica

CoefficientList[Series[1/((1-x)(1-4x)(1-9x)), {x, 0, 30}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)

PARI

Vec(1/((1-x)*(1-4*x)*(1-9*x))+O(x^30)) \\ Charles R Greathouse IV, Sep 23 2012

Sage

[(10 - 4^(n+4) +6*9^(n+2))/240 for n in (0..30)] # G. C. Greubel, Jul 04 2019

Formula

a(n) = 1/24 - 4^(n+2)/15 + 9^(n+2)/40. - Antonio Alberto Olivares, Feb 03 2010

a(n) = 13*a(n-1) - 36*a(n-2) + 1, n >= 2. - Vincenzo Librandi, Mar 23 2011