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A001949 Solutions of a fifth-order probability difference equation.

%I A001949 M1127 N0430 #85 Mar 28 2025 11:28:14
%S A001949 0,0,0,0,0,1,2,4,8,16,32,63,124,244,480,944,1856,3649,7174,14104,
%T A001949 27728,54512,107168,210687,414200,814296,1600864,3147216,6187264,
%U A001949 12163841,23913482,47012668,92424472,181701728,357216192,702268543,1380623604,2714234540
%N A001949 Solutions of a fifth-order probability difference equation.
%C A001949 This sequence is the case r = 5 in the solution to an r-th order probability difference equation that can be found in Eqs. (4) and (3) on p. 356 of Dunkel (1925). (Equation (3) follows equation (4) in the paper!) For r = 2, we get a shifted version of A000071. For r = 3, we get a shifted version of A008937. For r = 4, we get a shifted version of A107066. For r = 6, we get a shifted version of A172316. See also the table in A172119. - _Petros Hadjicostas_, Jun 15 2019
%D A001949 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001949 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001949 Vincenzo Librandi, <a href="/A001949/b001949.txt">Table of n, a(n) for n = 0..1000</a>
%H A001949 O. Dunkel, <a href="http://www.jstor.org/stable/2298801">Solutions of a probability difference equation</a>, Amer. Math. Monthly, 32 (1925), 354-370; see pp. 356 and 369.
%H A001949 T. Langley, J. Liese, and J. Remmel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Langley/langley2.html">Generating Functions for Wilf Equivalence Under Generalized Factor Order</a>, J. Int. Seq. 14 (2011), Article #11.4.2.
%H A001949 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001949 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001949 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,0,0,-1)
%F A001949 For n >= 6, a(n+1) = 2*a(n) - a(n-5).
%F A001949 G.f.: x^5 / ( (x-1)*(x^5 + x^4 + x^3 + x^2 + x - 1) ).
%F A001949 a(n) = Sum_{k=1..n-4} Sum_{j=0..floor((n-k-4)/5)} (-1)^j*binomial(n-5*j-5, k-1)*binomial(n-k-5*j-4, j). - _Vladimir Kruchinin_, Oct 19 2011
%F A001949 4*a(n) = A000322(n+1) - 1. - _R. J. Mathar_, Aug 16 2017
%F A001949 From _Petros Hadjicostas_, Jun 15 2019: (Start)
%F A001949 a(n) = 1 + a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) for n >= 5. (See Eq. (4) and the Theorem with r = 5 on p. 356 of Dunkel (1925).)
%F A001949 a(n) = T(n - 5, 5) for n >= 5, where T(n, k) = Sum_{j = 0..floor(n/(k+1))} (-1)^j * binomial(n - k*j, n - (k+1)*j) * 2^(n - (k+1)*j) for 0 <= k <= n. This is _Richard Choulet_'s formula in A172119.
%F A001949 (End)
%p A001949 A001949:=1/(z-1)/(z**5+z**4+z**3+z**2+z-1); # _Simon Plouffe_ in his 1992 dissertation
%t A001949 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 *)
%t A001949 LinearRecurrence[{2,0,0,0,0,-1},{0,0,0,0,0,1},40] (* _Harvey P. Dale_, Jan 17 2015 *)
%o A001949 (Maxima)
%o A001949 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 */
%o A001949 (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
%Y A001949 Column k = 1 of A141020 (with a different offset) and second main diagonal of A141021 (with no zeros).
%Y A001949 Column k = 5 of A172119.
%Y A001949 Partial sums of A001591.
%K A001949 nonn,easy
%O A001949 0,7
%A A001949 _N. J. A. Sloane_
%E A001949 Name edited by _Petros Hadjicostas_, Jun 15 2019