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A006261 a(n) = Sum_{k=0..5} C(n,k).
(Formerly M1126)
37
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. - 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}. - Reinhard Zumkeller, Jun 17 2009

a(n) is the maximal number of regions in 5-space formed by n-1 4-dimensional hypercubes. - Carl Schildkraut, May 26 2015

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

S. C. Chan, Letter to N. J. A. Sloane, Oct. 1975

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

R. K. Guy, Letter to N. J. A. Sloane

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

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

R. Zumkeller, Enumerations of Divisors

Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

FORMULA

a(n) = binomial(n+1, 5) + binomial(n+1, 3) + binomial(n+1, 1). - Len Smiley, Oct 20 2001

G.f.: (1 - 4*x + 7*x^2 - 6*x^3 + 3*x^4)/(1-x)^6. - 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. - Reinhard Zumkeller, Jun 17 2009

a(n) = a(n-1) + A000127(n-1). - Christian Schroeder, Jan 04 2016

EXAMPLE

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

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 range(1, 38)] # 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

(Python)

A006261_list, m = [], [1, -3, 4, -2, 1, 1]

for _ in range(10**2):

    A006261_list.append(m[-1])

    for i in range(5):

        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

(PARI) a(n)=sum(k=0, 5, binomial(n, k)) \\ Charles R Greathouse IV, Apr 08 2016

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

Sequence in context: A054043 A052396 A051040 * A290987 A145112 A062259

Adjacent sequences:  A006258 A006259 A006260 * A006262 A006263 A006264

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, based on a suggestion from S. C. Chan, Jun 10 1975

STATUS

approved

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Last modified November 26 04:03 EST 2022. Contains 358353 sequences. (Running on oeis4.)