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A005720 Quadrinomial coefficients.
(Formerly M4702)
3
1, 10, 44, 135, 336, 728, 1428, 2598, 4455, 7282, 11440, 17381, 25662, 36960, 52088, 72012, 97869, 130986, 172900, 225379, 290444, 370392, 467820, 585650, 727155, 895986, 1096200, 1332289, 1609210, 1932416, 2307888, 2742168, 3242393, 3816330, 4472412, 5219775 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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
FORMULA
a(n)= binomial(n+1, 3)*(n^3+15*n^2+86*n-120)/120, n >= 2.
G.f.: (x^2)*(1+3*x-5*x^2+2*x^3)/(1-x)^7. (numerator polynomial is N4(6, x) from A063421).
a(0)=1, a(1)=10, a(2)=44, a(3)=135, a(4)=336, a(5)=728, a(6)=1428, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Jun 23 2011
a(n) = binomial(n,2) + 7*binomial(n,3) + 10*binomial(n,4) + 5*binomial(n,5) + binomial(n,6) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
MAPLE
A005720:=-(1+3*z-5*z**2+2*z**3)/(z-1)**7; [Conjectured by Simon Plouffe in his 1992 dissertation.]
MATHEMATICA
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 10, 44, 135, 336, 728, 1428}, 40] (* or *) Table[Binomial[n+1, 3] (n^3+15n^2+86n-120)/120, {n, 2, 41}] (* Harvey P. Dale, Jun 23 2011 *)
PROG
(PARI) a(n)=(n^6 + 15*n^5 + 85*n^4 - 135*n^3 - 86*n^2 + 120*n)/720 \\ Charles R Greathouse IV, Jun 23 2011
CROSSREFS
a(n)= A008287(n, 6), n >= 2 (seventh column of quadrinomial coefficients).
Sequence in context: A008532 A085582 A058310 * A060326 A200448 A124852
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)