|
| |
|
|
A005718
|
|
Quadrinomial coefficients: C(2+n,n)+C(3+n,n)+C(4+n,n).
(Formerly M2918)
|
|
3
| |
|
|
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; 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 R. Janjic (agnus(AT)blic.net), 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. [From David Consiglio, Jr. (davecons2002(AT)yahoo.com), Oct 27 2008]
|
|
|
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
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 (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.)
|
|
|
MAPLE
| A005718:=-(3-3*z+z**2)/(z-1)**5; [Conjectured by S. 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}] (* From 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
| a(n)= A008287(n, 4), n >= 2 (fifth column of quadrinomial coefficients).
a(n)= A062745(n, 4), n >= 2 (fifth column).
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 (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Better description from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 02 2005
|
| |
|
|
