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Search: id:A005718
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| A005718 |
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Quadrinomial coefficients: C(2+n,n)+C(3+n,n)+C(4+n,n).
(Formerly M2918)
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+0
2
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| 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; listen)
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OFFSET
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0,1
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COMMENT
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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]
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
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LINKS
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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.
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FORMULA
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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) = 3*n + 6*(binomial(n,2))+4*(binomial(n,3))+binomial(n,4) for n >= 4. [From David Consiglio, Jr. (davecons2002(AT)yahoo.com), Oct 27 2008]
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EXAMPLE
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Example: a(2) = 12 and the 12 numbers 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 30, 31 all have a digit sum between 1 and 4 inclusive and contain no digits greater than 3. [From David Consiglio, Jr. (davecons2002(AT)yahoo.com), Oct 27 2008]
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MAPLE
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A005718:=-(3-3*z+z**2)/(z-1)**5; [Conjectured by S. Plouffe in his 1992 dissertation.]
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CROSSREFS
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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 this_sequence A098500 A037236 A054602
Adjacent sequences: A005715 A005716 A005717 this_sequence A005719 A005720 A005721
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Better description from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 02 2005
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