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A008860 a(n) = Sum_{k=0..7} binomial(n,k). 14
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; text; internal format)
OFFSET

0,2

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. - Geoffrey Critzer, Jan 19 2009

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

FORMULA

a(n) = Sum_{k=1..4} binomial(n+1, 2k-1) = (n^6 - 14*n^5 + 112*n^4 - 350*n^3 + 1099*n^2 + 364*n + 3828)*n/5040 + 1. [Len Smiley's formula for A006261, copied by Frank Ellermann]

G.f.: 1 - 6x + 16x^2 - 24x^3 + 22x^4 - 12x^5 + 4x^6/(1-x)^8. - Geoffrey Critzer, Jan 19 2009

EXAMPLE

a(8)=255 because there are 255 compositions of 9 into eight or fewer parts. - Geoffrey Critzer, Jan 23 2009

PROG

(Sage) [binomial(n, 1)+binomial(n, 3)+binomial(n, 5)+binomial(n, 7) for n in xrange(1, 36)] # Zerinvary Lajos, 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

(Haskell)

a008860 = sum . take 8 . a007318_row  -- Reinhard Zumkeller, Nov 24 2012

CROSSREFS

Cf. A008859, A008861, A008862, A008863, A006261, A000127.

Cf. A007318, A219531.

Sequence in context: A230177 A216264 A054045 * A145114 A172317 A234589

Adjacent sequences:  A008857 A008858 A008859 * A008861 A008862 A008863

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, R. K. Guy

STATUS

approved

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Last modified February 25 11:08 EST 2018. Contains 299653 sequences. (Running on oeis4.)