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A005585
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5-dimensional pyramidal numbers: n(n+1)(n+2)(n+3)(2n+3)/5!.
(Formerly M4387)
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19
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1, 7, 27, 77, 182, 378, 714, 1254, 2079, 3289, 5005, 7371, 10556, 14756, 20196, 27132, 35853, 46683, 59983, 76153, 95634, 118910, 146510, 179010, 217035, 261261, 312417, 371287, 438712, 515592, 602888, 701624, 812889, 937839
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Convolution of triangular numbers (A000217) and squares (A000290) (n>=1) - Graeme McRae (g_m(AT)mcraefamily.com), Jun 07 2006
p^k divides a(p^k-3), a(p^k-2), a(p^k-1) and a(p^k) for prime p>5 and integer k>0. p^k divides a((p^k-3)/2)) for prime p>5 and integer k>0. - Alexander Adamchuk (alex(AT)kolmogorov.com), May 08 2007
If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-5) is the number of 6-subsets of X intersecting both Y and Z. - Milan R. Janjic (agnus(AT)blic.net), Sep 08 2007
5-dimensional square numbers, fourth partial sums of binomial transform of [1,2,0,0,0,...]. a(n)=sum{i=0,n,C(n+4,i+4)*b(i)}, where b(i)=[1,2,0,0,0,...]. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009]
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 797.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Alexander Adamchuk (alex(AT)kolmogorov.com), May 08 2007, Table of n, a(n) for n = 1..121
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Milan Janjic, Two Enumerative Functions
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 entries for sequences related to Chebyshev polynomials.
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FORMULA
| G.f.: x*(1+x)/(1-x)^6.
a(n)=2*C(n+4, 5)-C(n+3, 4). - Paul Barry (pbarry(AT)wit.ie), Mar 04 2003
a(n)=C(n+3, 5)+C(n+4, 5). - Paul Barry (pbarry(AT)wit.ie), Mar 17 2003
binomial(n+2,6)-binomial(n,6), n>=4. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 21 2006
a(n) = Sum[ T(k)*T(k+1)/3, {k,1,n} ], where T(n) = n(n+1)/2 is a triangular number. - Alexander Adamchuk (alex(AT)kolmogorov.com), May 08 2007
a(n-1) = (1/4)*sum {1 <= x_1, x_2 <= n} |x_1*x_2*det V(x_1,x_2)| = (1/4)*sum {1 <= i,j <= n} i*j*|i-j|, where V(x_1,x_2} is the Vandermonde matrix of order 2. First differences of A040977. - Peter Bala (pbala(AT)toucansurf.com), Sep 21 2007
a(n)=C(n+4,4)+2*C(n+4,5) [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009]
a(1)=1, a(2)=7, a(3)=27, a(4)=77, a(5)=182, a(6)=378, a(n)=6*a(n-1)- 15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6) [From Harvey P. Dale, Oct 04 2011]
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MAPLE
| [seq(binomial(n+2, 6)-binomial(n, 6), n=4..45)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 21 2006
A005585:=(1+z)/(z-1)**6; [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| s1=s2=s3=0; lst={}; Do[s1+=n^2; s2+=s1; s3+=s2; AppendTo[lst, s3], {n, 0, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 15 2009]
With[{c=5!}, Table[n(n+1)(n+2)(n+3)(2n+3)/c, {n, 40}]] (* or *) LinearRecurrence[ {6, -15, 20, -15, 6, -1}, {1, 7, 27, 77, 182, 378}, 40] (* From Harvey P. Dale, Oct 04 2011 *)
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CROSSREFS
| a(n)= ((-1)^(n+1))*A053120(2*n+3, 5)/16 ( 1/16 of sixth unsigned column of Chebyshev T-triangle, zeros omitted).
Partial sums of A002415.
Cf. A006542, A040977, A047819.
Sequence in context: A162210 A161716 A162493 * A161410 A027180 A036597
Adjacent sequences: A005582 A005583 A005584 * A005586 A005587 A005588
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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