OFFSET
0,1
COMMENTS
In the Carlitz et al. reference a(n)= Q_{5,n+2}(2), n >= 0, with a(n)=binomial(11+n,n+2)-(n+3)*binomial(n+6,n+2), (eq.(3.3), p. 356, with n=5, m->n+2,r=2). Q_{5,m}(2) is the number of sequences (i_1,i_2,...,i_m) with i_s, s=1,...,m, from {1,2,3,4,5} (repetitions allowed), with exactly 2 increases between successive elements (first position is counted as an increase).
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
L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374, p. 351.
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).
FORMULA
a(n) = A035343(n+3, 9) = binomial(n+6, 6)*(n^3+42*n^2+677*n+5040)/(9!/6!).
G.f.: (10-20*x+15*x^2-4*x^3)/(1-x)^10; numerator polynomial is N5(9, x) from the array A063422.
a(n) = 10*C(n+3,3) + 40*C(n+3,4) + 65*C(n+3,5) + 56*C(n+3,6) + 28*C(n+3,7) + 8*C(n+3,8) + C(n+3,9) (see comment in A213887). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(n) = Sum_{k=1..10} (-1)^k * binomial(10,k) * a(n-k), a(0)=10. - G. C. Greubel, Aug 03 2015
a(n) = [x^9] (1+x+x^2+x^3+x^4)^(n+3). - Joerg Arndt, Aug 04 2015
MATHEMATICA
CoefficientList[Series[(10-20*x+15*x^2-4*x^3)/(1-x)^10, {x, 0, 50}], x](* Vincenzo Librandi, Mar 28 2012 *)
PROG
(PARI) a(n) = polcoeff((1+x+x^2+x^3+x^4)^(n+3), 9); \\ Joerg Arndt, Aug 04 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Comments and more terms from Wolfdieter Lang, Aug 29 2001
More terms from Sean A. Irvine, Nov 24 2010
STATUS
approved