OFFSET
0,2
COMMENTS
This is a general comment about sequences: A000012, A000027, A000124, A000125, A000127, A006261, A008859, this sequence, 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
a(n) is the maximal number of regions in 7-space formed by n-1 6-dimensional hypercubes. Also the number of binary words of length n matching the regular expression 1*0*1*0*1*0*1*0*. A000124, A000125, A000127, A006261, A008859 count binary words of the form 0*1*0*, 1*0*1*0*, 0*1*0*1*0*, 1*0*1*0*1*0*, and 0*1*0*1*0*1*0* respectively. - Manfred Scheucher, Jun 22 2023
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 and 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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved