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A000389 Binomial coefficients C(n,5).
(Formerly M4142 N1719)
77
0, 0, 0, 0, 0, 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, 53130, 65780, 80730, 98280, 118755, 142506, 169911, 201376, 237336, 278256, 324632, 376992, 435897, 501942, 575757, 658008, 749398 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Number of inequivalent ways of coloring the vertices of a regular 4-dimensional simplex with n colors, under the full symmetric group S_5 of order 120, with cycle index (x1^5 + 10*x1^3*x2 + 20*x1^2*x3 + 15*x1*x2^2 + 30*x1*x4 + 20*x2*x3 + 24*x5)/120.

Figurate numbers based on 5-dimensional regular simplex. According to Hyun Kwang Kim, it appears that every nonnegative integer can be represented as the sum of g = 10 of these 5-simplex(n) numbers (compared with g=3 for triangular numbers, g=5 for tetrahedral numbers and g=8 for pentatope numbers). - Jonathan Vos Post, Nov 28 2004

a(n) = -A110555(n+1,5). - Reinhard Zumkeller, Jul 27 2005

The convolution of the nonnegative integers (A001477) with the tetrahedral numbers (A000292), which themselves are the convolution of the nonnegative integers with themselves (making appropriate allowances for offsets of all sequences). - Graeme McRae, Jun 07 2006

a(n) is the number of terms in the expansion of (a_1+a_2+a_3+a_4+a_5+a_6)^n. - Sergio Falcon, Feb 12 2007

a(n)=A052787(n+5)/120. - Zerinvary Lajos, Apr 26 2007

Product of five consecutive numbers divided by 120. - Artur Jasinski, Dec 02 2007

Equals binomial transform of [1, 5, 10, 10, 5, 1, 0, 0, 0,...]. - Gary W. Adamson, Feb 02 2009

Equals INVERTi transform of A099242 (1, 7, 34, 153, 686, 3088,...). - Gary W. Adamson, Feb 02 2009

For a team with n basketball players (n>=5), this sequence is the number of possible starting lineups of 5 players, without regard to the positions (center, forward, guard) of the players. - Mohammad K. Azarian, Sep 10 2009

a(n) is the number of different patterns, regardless of order, when throwing (n-5) 6-sided dice. For example, one dice can display the 6 numbers 1, 2, ..., 6; two dice can display the 21 digit-pairs 11, 12, ..., 56, 66. - Ian Duff, Nov 16 2009

The sum of the first n pentatope numbers (1, 5, 15, 35, 70, 126, 210, ..., see A000332). - Paul Muljadi, Dec 16 2009

Sum{n=>0, a(n)/n!} = e/120. Sum{n=>4, a(n)/(n-4)!} = 501*e/120. See A067764 regarding the second ratio. - Richard R. Forberg, Dec 26 2013

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.

Gupta, Hansraj; Partitions of $j$-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).

J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

LINKS

T. D. Noe, Table of n, a(n) for n=0..1000

Milan Janjic, Two Enumerative Functions

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 255

H. K. Kim, On Regular Polytope Numbers, Journal: Proc. Amer.Math. Soc. 131 (2003), 65-75, as PDF file.

Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7

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.

J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.

Eric Weisstein's World of Mathematics, Composition.

Index entries for sequences related to linear recurrences with constant coefficients

FORMULA

G.f.: x^5/(1-x)^6.

a(n) = (n^5-10*n^4+35*n^3-50*n^2+24*n)/120. (Replace all x_i's in the cycle index by n.)

a(n+2) = sum_{i+j+k=n} i*j*k. - Benoit Cloitre, Nov 01 2002

Convolution of triangular numbers (A000217) with themselves.

Partial sums of A000332. - Alexander Adamchuk, Dec 19 2004

a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)/120.

a(n+3) = (1/2!)*diff(S(n,x),x$2)|_{x=2}, n>=2, One half of second derivative of Chebyshev S-polynomials evaluated at x=2. See A049310. - Wolfdieter Lang, Apr 04 2007

sum(n>=5, 1/a(n) = 5/4. - R. J. Mathar, Jan 27 2009

For n>4 a(n)=1/(Integral_{x=0..Pi/2} 10*(sin(x))^(2*n-9)*(cos(x))^9). - Francesco Daddi, Aug 02 2011

MAPLE

f:=n->(1/120)*(n^5-10*n^4+35*n^3-50*n^2+24*n);

ZL := [S, {S=Prod(B, B, B, B, B, B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=6..42); - Zerinvary Lajos, Mar 13 2007

A000389:=1/(z-1)**6; [Simon Plouffe, 1992 dissertation.]

MATHEMATICA

Table[Binomial[n, 5], {n, 5, 50}] - Stefan Steinerberger, Apr 02 2006

PROG

(PARI) conv(u, v)=local(w); w=vector(length(u), i, sum(j=1, i, u[j]*v[i+1-j])); w; t(n)=n*(n+1)/2; u=vector(10, i, t(i)); conv(u, u)

(Haskell)

a000389 n | n <= 2    = 0

          | otherwise = sum $ zipWith (*) ts $ reverse ts where

                        ts = take (n - 2) a000217_list

-- Reinhard Zumkeller, Apr 13 2012

CROSSREFS

Cf. A002299, A053127, A000332, A000579, A000580, A000581, A000582.

Cf. A000217, A005583, A051747, A000292, A000332.

Cf. A099242. - Gary W. Adamson, Feb 02 2009

Sequence in context: A120478 A023031 A090581 * A143980 A140228 A006090

Adjacent sequences:  A000386 A000387 A000388 * A000390 A000391 A000392

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000

Corrected formulas that had been based on other offsets. - R. J. Mathar, Jun 16 2009

I changed the offset to 0. This will require some further adjustments to the formulas. - N. J. A. Sloane, Aug 01 2010

STATUS

approved

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Last modified April 18 11:35 EDT 2014. Contains 240707 sequences.