A001919 Eighth column of quadrinomial coefficients.

6, 40, 155, 456, 1128, 2472, 4950, 9240, 16302, 27456, 44473, 69680, 106080, 157488, 228684, 325584, 455430, 627000, 850839, 1139512, 1507880, 1973400, 2556450, 3280680, 4173390, 5265936, 6594165, 8198880, 10126336, 12428768, 15164952, 18400800, 22209990

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
• 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

• T. D. Noe, Table of n, a(n) for n = 3..1000
• L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374.
• 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; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Programs

Maple

seq(n*(n^2-1)*(n^2-4)*(n^2+21*n+180)/5040,n=3..34); # Emeric Deutsch, Jan 27 2005
A001919:=(3*z**2-8*z+6)/(z-1)**8; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Table[n*(n^2 - 1)*(n^2 - 4)*(n^2 + 21*n + 180)/5040, {n, 3, 50}] (* T. D. Noe, Aug 17 2012 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{6,40,155,456,1128,2472,4950,9240},40] (* Harvey P. Dale, Mar 27 2013 *)

Formula

a(n) = A008287(n, 7) = binomial(n+2, 5)*(n^2+21*n+180 )/42, n >= 3.

G.f.: (x^3)*(6-8*x+3*x^2 )/(1-x)^8. Numerator polynomial is N4(7, x) from array A063421.

a(n) = n(n^2-1)(n^2-4)(n^2+21n+180)/5040. - Emeric Deutsch, Jan 27 2005

a(n) = 6*C(n,3) + 16*C(n,4) + 15*C(n,5) + 6*C(n,6) + C(n,7) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

a(3)=6, a(4)=40, a(5)=155, a(6)=456, a(7)=1128, a(8)=2472, a(9)=4950, a(10)=9240, a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)- 28*a(n-6)+ 8*a(n-7)-a(n-8). - Harvey P. Dale, Mar 27 2013

Extensions

More terms from Emeric Deutsch, Jan 27 2005