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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 Sum_{i=0..j-1} x^i/(1-x)^(i+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

Ângela Mestre, José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.

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 - 6*x + 16*x^2 - 24*x^3 + 22*x^4 - 12*x^5 + 4*x^6)/(1-x)^8. - Geoffrey Critzer, Jan 19 2009 [Corrected by Georg Fischer, May 19 2019]

EXAMPLE

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

MAPLE

seq(sum(binomial(n, j), j=0..7), n=0..40); # G. C. Greubel, Sep 13 2019

MATHEMATICA

CoefficientList[Series[(1-6x+16x^2-24x^3+22x^4-12x^5+4x^6)/(1-x)^8, {x, 0, 34}], x] (* Georg Fischer, May 19 2019 *)

Sum[Binomial[Range[41]-1, j-1], {j, 8}] (* G. C. Greubel, Sep 13 2019 *)

PROG

(Sage) [binomial(n, 1)+binomial(n, 3)+binomial(n, 5)+binomial(n, 7) for n in range(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

(MAGMA) [&+[Binomial(n, k): k in [0..7]]: n in [0..55]]; // Vincenzo Librandi, May 20 2019

(Sage) [sum(binomial(n, k) for k in (0..7)) for n in (0..40)] # G. C. Greubel, Sep 13 2019

(GAP) List([0..40], n-> Sum([0..7], k-> Binomial(n, k)) ); # G. C. Greubel, Sep 13 2019

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 October 25 15:49 EDT 2020. Contains 338012 sequences. (Running on oeis4.)