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A006261
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Sum_{ k = 0..5 } C(n,k).
(Formerly M1126)
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30
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1, 2, 4, 8, 16, 32, 63, 120, 219, 382, 638, 1024, 1586, 2380, 3473, 4944, 6885, 9402, 12616, 16664, 21700, 27896, 35443, 44552, 55455, 68406, 83682, 101584, 122438, 146596, 174437, 206368, 242825, 284274, 331212, 384168, 443704
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n) is the sum of the first six terms of the n-th row in Pascal's triangle. [From Geoffrey Critzer, Jan 19 2009]
Also the interpolating polynomial for the divisors of 32: {a(k):0<=k<6}={1,2,4,8,16,32}. [From Reinhard Zumkeller, Jun 17 2009]
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
M. L. Cornelius, Variations on a geometric progression, Mathematics in School, 4 (No. 3, May 1975), p. 32.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
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.
R. Zumkeller, Enumerations of Divisors [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]
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FORMULA
| a(n)=binomial(n+1, 5)+binomial(n+1, 3)+binomial(n+1, 1). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Oct 20 2001
G.f.: (1-4*x+7*x^2-6*x^3+3*x^4)/(1-x)^6 [From Geoffrey Critzer, Jan 19 2009]
E.g.f.: (1+x+x^2/2+x^3/6+x^4/24+x^5/120)*exp(x)
a(n) = (n^5 - 5*n^4 + 25*n^3 + 5*n^2 + 94*n + 120)/120. [From Reinhard Zumkeller, Jun 17 2009]
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EXAMPLE
| a(7)=120 because the first six terms in the 7th row of Pascal's triangle 1+7+21+35+35+21=120 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jan 19 2009]
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MAPLE
| A006261:=(z**2-z+1)*(3*z**2-3*z+1)/(z-1)**6; [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| CoefficientList[
Series[(1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120) Exp[x], {x, 0,
52}], x]*Table[n!, {n, 0, 52}]
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PROG
| (Sage) [binomial(n, 1)+binomial(n, 3)+binomial(n, 5) for n in xrange(1, 38)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]
(MAGMA) [(n^5 - 5*n^4 + 25*n^3 + 5*n^2 + 94*n + 120)/120: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011
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CROSSREFS
| A057703(n) + 1.
A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715. [From Reinhard Zumkeller, Jun 17 2009]
Sequence in context: A054043 A052396 A051040 * A145112 A062259 A001949
Adjacent sequences: A006258 A006259 A006260 * A006262 A006263 A006264
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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