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A001949 A probability difference equation.
(Formerly M1127 N0430)
9
0, 0, 0, 0, 0, 1, 2, 4, 8, 16, 32, 63, 124, 244, 480, 944, 1856, 3649, 7174, 14104, 27728, 54512, 107168, 210687, 414200, 814296, 1600864, 3147216, 6187264, 12163841, 23913482, 47012668, 92424472, 181701728, 357216192, 702268543, 1380623604, 2714234540 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

REFERENCES

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).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

O. Dunkel, Solutions of a probability difference equation, Amer. Math. Monthly, 32 (1925), 354-370.

T. Langley, J. Liese, J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order , J. Int. Seq. 14 (2011) # 11.4.2

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,0,-1)

FORMULA

For n >= 6, a(n+1) = 2*a(n)-a(n-5).

G.f.: x^5 / ( (x-1)*(x^5+x^4+x^3+x^2+x-1) ).

a(n) = sum(k=1..n-4, sum(j=0..(n-k-4)/5, (-1)^j*binomial(n-5*j-5,k-1)*binomial(n-k-5*j-4,j),j,0,(n-k-4)/5)). [Vladimir Kruchinin, Oct 19 2011]

4*a(n) = A000322(n+1) -1. - R. J. Mathar, Aug 16 2017

MAPLE

A001949:=1/(z-1)/(z**5+z**4+z**3+z**2+z-1); # Simon Plouffe in his 1992 dissertation

MATHEMATICA

t={0, 0, 0, 0, 0}; Do[AppendTo[t, t[[-5]]+t[[-4]]+t[[-3]]+t[[-2]]+t[[-1]]+1], {n, 40}]; t (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)

LinearRecurrence[{2, 0, 0, 0, 0, -1}, {0, 0, 0, 0, 0, 1}, 40] (* Harvey P. Dale, Jan 17 2015 *)

PROG

(Maxima)

a(n):=sum(sum((-1)^j*binomial(n-5*j-5, k-1)*binomial(n-k-5*j-4, j), j, 0, (n-k-4)/5), k, 1, n-4); /* Vladimir Kruchinin, Oct 19 2011 */

(PARI) x='x+O('x^99); concat(vector(5), Vec(x^5/((x-1)*(x^5+x^4+x^3+x^2+x-1)))) \\ Altug Alkan, Oct 04 2017

CROSSREFS

Sequence in context: A290987 A145112 A062259 * A210031 A239558 A239559

Adjacent sequences:  A001946 A001947 A001948 * A001950 A001951 A001952

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 18 17:18 EDT 2019. Contains 322229 sequences. (Running on oeis4.)