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%I M1126 #98 Jun 24 2023 12:44:24
%S 1,2,4,8,16,32,63,120,219,382,638,1024,1586,2380,3473,4944,6885,9402,
%T 12616,16664,21700,27896,35443,44552,55455,68406,83682,101584,122438,
%U 146596,174437,206368,242825,284274,331212,384168,443704
%N a(n) = Sum_{k=0..5} binomial(n,k).
%C a(n) is the sum of the first six terms of the n-th row in Pascal's triangle. - _Geoffrey Critzer_, Jan 19 2009
%C Also the interpolating polynomial for the divisors of 32: {a(k): 0 <= k < 6} = {1,2,4,8,16,32}. - _Reinhard Zumkeller_, Jun 17 2009
%C a(n) is the maximal number of regions in 5-space formed by n-1 4-dimensional hypercubes. - _Carl Schildkraut_, May 26 2015
%C a(n) is the number of binary words of length n matching the regular expression 1*0*1*0*1*0*. A000124, A000125, A000127 count binary words of the form 0*1*0*, 1*0*1*0*, and 0*1*0*1*0*, respectively. - _Manfred Scheucher_, Jun 22 2023
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
%D M. L. Cornelius, Variations on a geometric progression, Mathematics in School, 4 (No. 3, May 1975), p. 32.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A006261/b006261.txt">Table of n, a(n) for n = 0..10000</a>
%H S. C. Chan, <a href="/A006261/a006261.pdf">Letter to N. J. A. Sloane, Oct. 1975</a>
%H M. L. Cornelius, <a href="/A006261/a006261_1.pdf">Variations on a geometric progression</a>, Mathematics in School, 4 (No. 3, May 1975), p. 32. (Annotated scanned copy)
%H R. K. Guy, <a href="/A000346/a000346.pdf">Letter to N. J. A. Sloane</a>
%H Ângela Mestre, 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 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H R. Zumkeller, <a href="/A161700/a161700.txt">Enumerations of Divisors</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6, -15, 20, -15, 6, -1).
%F a(n) = binomial(n+1, 5) + binomial(n+1, 3) + binomial(n+1, 1). - _Len Smiley_, Oct 20 2001
%F G.f.: (1 - 4*x + 7*x^2 - 6*x^3 + 3*x^4)/(1-x)^6. - _Geoffrey Critzer_, Jan 19 2009
%F E.g.f.: (1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120)*exp(x)
%F a(n) = (n^5 - 5*n^4 + 25*n^3 + 5*n^2 + 94*n + 120)/120. - _Reinhard Zumkeller_, Jun 17 2009
%F a(n) = a(n-1) + A000127(n-1). - _Christian Schroeder_, Jan 04 2016
%e a(7) = 120 because the first six terms in the 7th row of Pascal's triangle 1 + 7 + 21 + 35 + 35 + 21 = 120. - _Geoffrey Critzer_, Jan 19 2009
%p A006261:=(z**2-z+1)*(3*z**2-3*z+1)/(z-1)**6; # _Simon Plouffe_ in his 1992 dissertation
%t CoefficientList[
%t Series[(1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120) Exp[x], {x, 0,
%t 52}], x]*Table[n!, {n, 0, 52}]
%o (Sage) [binomial(n,1)+binomial(n,3)+binomial(n,5) for n in range(1, 38)] # _Zerinvary Lajos_, May 17 2009
%o (Magma) [(n^5 - 5*n^4 + 25*n^3 + 5*n^2 + 94*n + 120)/120: n in [0..40]]; // _Vincenzo Librandi_, Jul 17 2011
%o (Haskell)
%o a006261 = sum . take 6 . a007318_row -- _Reinhard Zumkeller_, Nov 24 2012
%o (Python)
%o A006261_list, m = [], [1, -3, 4, -2, 1, 1]
%o for _ in range(10**2):
%o A006261_list.append(m[-1])
%o for i in range(5):
%o m[i+1] += m[i] # _Chai Wah Wu_, Jan 24 2016
%o (PARI) a(n)=sum(k=0,5,binomial(n,k)) \\ _Charles R Greathouse IV_, Apr 08 2016
%Y A057703(n) + 1.
%Y A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A007318, A008859, A008860, A008861, A008862, A008863, A219531.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, based on a suggestion from S. C. Chan, Jun 10 1975
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