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A008860
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Sum C(n,k), k=0..7.
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9
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1, 2, 4, 8, 16, 32, 64, 128, 255, 502, 968, 1816, 3302, 5812, 9908, 16384, 26333, 41226, 63004, 94184, 137980, 198440, 280600, 390656, 536155, 726206, 971712, 1285624, 1683218, 2182396, 2804012, 3572224, 4514873, 5663890, 7055732
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| This is a general comment about sequences: A000012, A000027, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863. Let j in {1, 2, ...11} index these 11 sequences respective to their order above. Then a(n) in each sequence is the number of compositions of (n+1) into j or fewer parts. From this we see that the ordinary generating function for each sequence is the Sum x^i/(1-x)^(i+1), i=0, j-1. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jan 19 2009]
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
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FORMULA
| a(n) = sum( binomial( n+1, 2k-1 ) for k=1..4 ) = ( n^6 -14*n^5 +112*n^4 -350*n^3 +1099*n^2 +364*n +3828 )*n/5040 +1.
G.f.:1-6x+16x^2-24x^3+22x^4-12x^5+4x^6/(1-x)^8 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jan 19 2009]
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EXAMPLE
| a(8)=255 because there are 255 compositions of 9 into eight or fewer parts. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jan 23 2009]
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PROG
| (Other) sage: [binomial(n, 1)+binomial(n, 3)+binomial(n, 5)+binomial(n, 7) for n in xrange(1, 36)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]
(PARI) a(n)=(n+1)*(n^6-15*n^5+127*n^4-477*n^3+1576*n^2-1212*n+5040)/5040 \\ Charles R Greathouse IV, Dec 07 2011
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CROSSREFS
| Cf. A008859, A008861, A008862, A008863, A006261, A000127.
Sequence in context: A009641 A089889 A054045 * A145114 A172317 A079262
Adjacent sequences: A008857 A008858 A008859 * A008861 A008862 A008863
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
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EXTENSIONS
| Len Smiley's formula for A006261 copied by Frank Ellermann (hmdmhdfmhdjmzdtjmzdtzktdkztdjz(AT)gmail.com).
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