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A008861 a(n) = Sum_{k=0..8} C(n,k). 14
1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1013, 1981, 3797, 7099, 12911, 22819, 39203, 65536, 106762, 169766, 263950, 401930, 600370, 880970, 1271626, 1807781, 2533987, 3505699, 4791323, 6474541, 8656937, 11460949, 15033173, 19548046 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is the number of compositions (ordered partitions) of n+1 into nine or fewer parts. - Geoffrey Critzer, Jan 24 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 (9,-36,84,-126,126,-84,36,-9,1).

FORMULA

a(n) = Sum_{k=0..4} binomial(n+1, 2*k), compare A008859.

From Geoffrey Critzer, Jan 24 2009: (Start)

G.f.: (1 -7*x +22*x^2 -40*x^3 +46*x^4 -34*x^5 +16*x^6 -4*x^7 +x^8)/(1-x)^9.

a(n) = (n^8 -20*n^7 +210*n^6 -1064*n^5 +3969*n^4 -4340*n^3 +15980*n^2 +25584*n +40320)/8!. (End)

EXAMPLE

a(9)=511 because all but one (namely 1+1+1+...+1=10) of the 2^9 compositions of 10 are in nine or fewer parts. - Geoffrey Critzer, Jan 24 2009

MAPLE

seq(sum(binomial(n, j), j=0..8), n=0..40); # G. C. Greubel, Sep 13 2019

MATHEMATICA

Sum[Binomial[Range[41]-1, j-1], {j, 9}] (* G. C. Greubel, Sep 13 2019 *)

PROG

(Haskell)

a008861 = sum . take 9 . a007318_row -- Reinhard Zumkeller, Nov 24 2012

(PARI) vector(40, n, sum(j=0, 8, binomial(n-1, j))) \\ G. C. Greubel, Sep 13 2019

(Magma) [(&+[Binomial(n, k): k in [0..8]]): n in [0..40]]; // G. C. Greubel, Sep 13 2019

(Sage) [sum(binomial(n, k) for k in (0..8)) for n in (0..40)] # G. C. Greubel, Sep 13 2019

(GAP) List([0..40], n-> Sum([0..8], k-> Binomial(n, k)) ); # G. C. Greubel, Sep 13 2019

CROSSREFS

Cf. A008859, A008860, A008862, A008863, A006261, A000127.

Cf. A007318, A219531.

Sequence in context: A271481 A208849 A054046 * A145115 A172318 A234590

Adjacent sequences: A008858 A008859 A008860 * A008862 A008863 A008864

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified November 29 16:39 EST 2022. Contains 358431 sequences. (Running on oeis4.)