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Search: id:A008860
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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
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OFFSET
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0,2
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COMMENT
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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
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
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FORMULA
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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
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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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PROGRAM
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(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]
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CROSSREFS
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Cf. A008859, A008861, A008862, A008863, A006261, A000127.
Sequence in context: A009641 A089889 A054045 this_sequence A145114 A079262 A087079
Adjacent sequences: A008857 A008858 A008859 this_sequence A008861 A008862 A008863
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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), R. K. Guy
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EXTENSIONS
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Len Smiley's formula for A006261 copied by frank.ellermann(AT)t-online.de
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