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%I #73 Jun 24 2023 13:24:42
%S 1,2,4,8,16,32,64,128,255,502,968,1816,3302,5812,9908,16384,26333,
%T 41226,63004,94184,137980,198440,280600,390656,536155,726206,971712,
%U 1285624,1683218,2182396,2804012,3572224,4514873,5663890,7055732
%N a(n) = Sum_{k=0..7} binomial(n,k).
%C 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
%C 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
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
%H Reinhard Zumkeller, <a href="/A008860/b008860.txt">Table of n, a(n) for n = 0..10000</a>
%H Ângela Mestre and José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F 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_]
%F 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]
%e a(8)=255 because there are 255 compositions of 9 into eight or fewer parts. - _Geoffrey Critzer_, Jan 23 2009
%p seq(sum(binomial(n,j), j=0..7), n=0..40); # _G. C. Greubel_, Sep 13 2019
%t 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 *)
%t Sum[Binomial[Range[41]-1, j-1], {j,8}] (* _G. C. Greubel_, Sep 13 2019 *)
%o (Sage) [binomial(n,1)+binomial(n,3)+binomial(n,5)+binomial(n,7) for n in range(1, 36)] # _Zerinvary Lajos_, May 17 2009
%o (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
%o (Haskell)
%o a008860 = sum . take 8 . a007318_row -- _Reinhard Zumkeller_, Nov 24 2012
%o (Magma) [&+[Binomial(n, k): k in [0..7]]: n in [0..55]]; // _Vincenzo Librandi_, May 20 2019
%o (Sage) [sum(binomial(n,k) for k in (0..7)) for n in (0..40)] # _G. C. Greubel_, Sep 13 2019
%o (GAP) List([0..40], n-> Sum([0..7], k-> Binomial(n,k)) ); # _G. C. Greubel_, Sep 13 2019
%Y Cf. A008859, A008861, A008862, A008863, A006261, A000127.
%Y Cf. A007318, A219531.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_ and _R. K. Guy_
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