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A006261 Sum_{ k = 0..5 } C(n,k).
(Formerly M1126)
35
1, 2, 4, 8, 16, 32, 63, 120, 219, 382, 638, 1024, 1586, 2380, 3473, 4944, 6885, 9402, 12616, 16664, 21700, 27896, 35443, 44552, 55455, 68406, 83682, 101584, 122438, 146596, 174437, 206368, 242825, 284274, 331212, 384168, 443704 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is the sum of the first six terms of the n-th row in Pascal's triangle. [From Geoffrey Critzer, Jan 19 2009]

Also the interpolating polynomial for the divisors of 32: {a(k):0<=k<6}={1,2,4,8,16,32}. [From Reinhard Zumkeller, Jun 17 2009]

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.

M. L. Cornelius, Variations on a geometric progression, Mathematics in School, 4 (No. 3, May 1975), p. 32.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

R. Zumkeller, Enumerations of Divisors [From Reinhard Zumkeller, Jun 17 2009]

FORMULA

a(n)=binomial(n+1, 5)+binomial(n+1, 3)+binomial(n+1, 1). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Oct 20 2001

G.f.: (1-4*x+7*x^2-6*x^3+3*x^4)/(1-x)^6 [From Geoffrey Critzer, Jan 19 2009]

E.g.f.: (1+x+x^2/2+x^3/6+x^4/24+x^5/120)*exp(x)

a(n) = (n^5 - 5*n^4 + 25*n^3 + 5*n^2 + 94*n + 120)/120. [From Reinhard Zumkeller, Jun 17 2009]

EXAMPLE

a(7)=120 because the first six terms in the 7th row of Pascal's triangle 1+7+21+35+35+21=120 [From Geoffrey Critzer, Jan 19 2009]

MAPLE

A006261:=(z**2-z+1)*(3*z**2-3*z+1)/(z-1)**6; [Simon Plouffe in his 1992 dissertation.]

MATHEMATICA

CoefficientList[

  Series[(1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120) Exp[x], {x, 0,

    52}], x]*Table[n!, {n, 0, 52}]

PROG

(Sage) [binomial(n, 1)+binomial(n, 3)+binomial(n, 5) for n in xrange(1, 38)]# [From Zerinvary Lajos, May 17 2009]

(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

(Haskell)

a006261 = sum . take 6 . a007318_row  -- Reinhard Zumkeller, Nov 24 2012

CROSSREFS

A057703(n) + 1.

A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715. [From Reinhard Zumkeller, Jun 17 2009]

Cf. A007318, A008859, A008860, A008861, A008862, A008863, A219531.

Sequence in context: A054043 A052396 A051040 * A145112 A062259 A001949

Adjacent sequences:  A006258 A006259 A006260 * A006262 A006263 A006264

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified July 24 08:51 EDT 2014. Contains 244878 sequences.