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A008778
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a(n) = (n+1)*(n^2 +8*n +6)/6. Number of n-dimensional partitions of 4. Number of terms in 4th derivative of a function composed with itself n times.
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30
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1, 5, 13, 26, 45, 71, 105, 148, 201, 265, 341, 430, 533, 651, 785, 936, 1105, 1293, 1501, 1730, 1981, 2255, 2553, 2876, 3225, 3601, 4005, 4438, 4901, 5395, 5921, 6480, 7073, 7701, 8365, 9066, 9805, 10583, 11401, 12260, 13161, 14105, 15093, 16126, 17205, 18331
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OFFSET
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0,2
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COMMENTS
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Let m(i,1)=i; m(1,j)=j; m(i,j)=m(i-1,j)-m(i-1,j-1); then a(n)=m(n+3,3) -Benoit Cloitre,May 08 2002
a(n) = number of (n+6)-bit binary sequences with exactly 6 1's none of which is isolated. -David Callan,Jul 15 2004
If a 2-set Y and 2-set Z, having one element in common, are subsets of an n-set X then a(n-4) is the number of 4-subsets of X intersecting both Y and Z. -Milan Janjic,Oct 03 2007
a(n) = Sum of first (n+1) triangular numbers plus n-th triangular number (see penultimate formula by Henry Bottomley). -Vladimir Joseph Stephan Orlovsky,Oct 13 2009
For n > 0, a(n-1) is the number of compositions of n+6 into n parts avoiding the part 2. -Milan Janjic,Jan 07 2016
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REFERENCES
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G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 190 eq. (11.4.7).
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LINKS
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FORMULA
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a(n) = dot_product(n, n-1,...2, 1)*(2, 3,..., n, 1) for n = 2, 3, 4,... [i.e., a(2) = (2, 1)*(2, 1), a(3) = (3, 2, 1)*(2, 3, 1)]. -Clark Kimberling
a(n) = Sum_{0<=k, l<=n; k+l|n} k*l. -Ralf Stephan,May 06 2005
E.g.f.: (6 +24*x +12*x^2 +x^3)*exp(x)/6. -G. C. Greubel,Sep 11 2019
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EXAMPLE
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G.f. = 1 + 5*x + 13*x^2 + 26*x^3 + 45*x^4 + 71*x^5 + 105*x^6 + 148*x^7 + 201*x^8 +...
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MAPLE
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seq(1+4*k+4*binomial(k, 2)+binomial(k, 3), k=0..45);
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MATHEMATICA
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LinearRecurrence[{4, -6, 4, -1}, {1, 5, 13, 26}, 51] (*G. C. Greubel,Sep 11 2019 *)
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PROG
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(PARI) Vec((1+x-x^2)/(1-x)^4 + O(x^50)) \\Altug Alkan,Jan 07 2016
(Sage) [(n+1)*(n^2 +8*n +6)/6 for n in (0..50)] #G. C. Greubel,Sep 11 2019
(GAP) List([0..50], n-> (n+1)*(n^2 +8*n +6)/6); #G. C. Greubel,Sep 11 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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