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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

Table of n, a(n) for n=2..37.

R. K. Guy, Letter to N. J. A. Sloane, 1987

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 (7,-21,35,-35,21,-7,1).

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

Adjacent sequences:  A005717 A005718 A005719 * A005721 A005722 A005723

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 18 07:58 EDT 2018. Contains 316307 sequences. (Running on oeis4.)