|
| |
|
|
A000389
|
|
Binomial coefficients C(n,5).
(Formerly M4142 N1719)
|
|
67
| |
|
|
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; 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 (sfalcon(AT)dma.ulpgc.es), Feb 12 2007
a(n)=A052787(n+5)/120. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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,...]. [From Gary W. Adamson, Feb 02 2009]
Equals INVERTi transform of A099242 (1, 7, 34, 153, 686, 3088,...). [From 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, gaurd) of the players. [From 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 die can display the 6 numbers 1, 2, ..., 6; two dice can display the 21 digit-pairs 11, 12, ..., 56, 66. [From Ian Duff (ianfduff(AT)yahoo.co.uk), Nov 16 2009]
The sum of the first n pentatope numbers (1, 5, 15, 35, 70, 126, 210, ..., see A000332). [From Paul Muljadi, Dec 16 2009]
|
|
|
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, Jounal: 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
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for sequences related to linear recurrences with constant coefficients
|
|
|
FORMULA
| G.f.: x^5/(1-x)^6.
a(n) = (1/120)*(n^5-10*n^4+35*n^3-50*n^2+24*n). (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(n) - 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. W. Lang, Apr 04 2007.
sum(n>=5, 1/a(n) = 5/4. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 27 2009
For n>4 a(n)=1/(Integral_{x=0..Pi/2} 10*(sin(x))^(2*n-9)*(cos(x))^9). [From 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 (zerinvarylajos(AT)yahoo.com), Mar 13 2007
A000389:=1/(z-1)**6; [S. 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)
|
|
|
CROSSREFS
| Cf. A002299, A053127, A000332, A000579, A000580, A000581, A000582.
Cf. A000217, A005583, A051747, A000292, A000332.
Cf. A099242 [From 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 (njas(AT)research.att.com).
|
|
|
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 (mathar(AT)strw.leidenuniv.nl), 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
|
| |
|
|
