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A005720
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Quadrinomial coefficients.
(Formerly M4702)
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1
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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; internal format)
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OFFSET
| 2,2
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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).
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LINKS
| S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index to sequences with linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
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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) [From Harvey P. Dale, June 23 2011]
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MAPLE
| A005720:=-(1+3*z-5*z**2+2*z**3)/(z-1)**7; [Conjectured by S. Plouffe in his 1992 dissertation.]
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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}] (* From Harvey P. Dale, June 23 2011 *)
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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
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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
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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