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A005718 Quadrinomial coefficients: C(2+n,n) + C(3+n,n) + C(4+n,n).
(Formerly M2918)
7
3, 12, 31, 65, 120, 203, 322, 486, 705, 990, 1353, 1807, 2366, 3045, 3860, 4828, 5967, 7296, 8835, 10605, 12628, 14927, 17526, 20450, 23725, 27378, 31437, 35931, 40890, 46345, 52328, 58872, 66011, 73780, 82215, 91353, 101232, 111891, 123370, 135710, 148953, 163142, 178321, 194535 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
If Y is an (n-3)-subset of an n-set X then, for n>=5, a(n-5) is the number of 4-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007
This equation represents the number of numbers with <=n digits such that the sum of the digits is between 1 and 4 inclusive and no digit is larger than 3. - David Consiglio, Jr., Oct 27 2008
Row 2 of the convolution array A213548. - Clark Kimberling, Jun 20 2012
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, 2)*(n^2+7*n+18)/12, n >= 2.
G.f.: (x^2)*(3-3*x+x^2)/(1-x)^5. (numerator polynomial is N4(4, x) from A063421).
a(n) = A008287(n, 4), n >= 2 (fifth column of quadrinomial coefficients).
a(n) = A062745(n, 4), n >= 2 (fifth column).
a(n) = 3*C(n+2,2) + 3*C(n+2,3) + C(n+2,4) (see comment in A071675). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
MAPLE
A005718:=-(3-3*z+z**2)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Table[n (n + 1)/2! + n (n + 1) (n + 2)/3! + n (n + 1) (n + 2) (n + 3)/4!, {n, 1, 60}] (* Vladimir Joseph Stephan Orlovsky, Jun 14 2011 *)
Table[Plus@@Table[Binomial[i + n, n], {i, 2, 4}], {n, 0, 43}] (* From Alonso del Arte, Jun 14 2011 *)
PROG
(PARI) a(n)=(((n+14)*n+71)*n+130)*n/24+3 \\ Charles R Greathouse IV, Jun 14 2011
(Magma) [(((n+14)*n+71)*n+130)*n/24+3: n in [0..45]]; // Vincenzo Librandi, Jun 15 2011
CROSSREFS
Sequence in context: A009135 A131740 A037237 * A199231 A098500 A037236
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Better description from Zerinvary Lajos, Dec 02 2005
STATUS
approved

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