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A005718 Quadrinomial coefficients: C(2+n,n) + C(3+n,n) + C(4+n,n).
(Formerly M2918)
6
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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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 (5,-10,10,-5,1).

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

Adjacent sequences:  A005715 A005716 A005717 * A005719 A005720 A005721

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Better description from Zerinvary Lajos, Dec 02 2005

STATUS

approved

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Last modified February 20 20:12 EST 2018. Contains 299385 sequences. (Running on oeis4.)