A051747 -id:A051747 - cp's OEIS frontend

A000389 Binomial coefficients C(n,5).

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

Offset: 0

N. J. A. Sloane

a(n+4) is the 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
The convolution of the nonnegative integers (A001477) with the tetrahedral numbers (A000292), which 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
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 die 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
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
For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 4 elements, which is 3*C(n+1,5) (for n>=4), hence a(n) = 3*C(n+1,5) = 3*A000389(n+1). - Serhat Bulut, Mar 11 2015
a(n) = fallfac(n,5)/5! is also the number of independent components of an antisymmetric tensor of rank 5 and dimension n >= 1. Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
Number of compositions (ordered partitions) of n+1 into exactly 6 parts. - Juergen Will, Jan 02 2016
Number of weak compositions (ordered weak partitions) of n-5 into exactly 6 parts. - Juergen Will, Jan 02 2016
a(n+3) could be the general number of all geodetic graphs of diameter n>=2 homeomorphic to the Petersen Graph. - Carlos Enrique Frasser, May 24 2018
From Robert A. Russell, Dec 24 2020: (Start)
a(n) is the number of chiral pairs of colorings of the 5 tetrahedral facets (or vertices) of the regular 4-D simplex (5-cell, pentachoron, Schläfli symbol {3,3,3}) using subsets of a set of n colors. Each member of a chiral pair is a reflection but not a rotation of the other.
a(n+4) is the number of unoriented colorings of the 5 tetrahedral facets of the regular 4-D simplex (5-cell, pentachoron) using subsets of a set of n colors. Each chiral pair is counted as one when enumerating unoriented arrangements. (End)
For integer m and positive integer r >= 4, the polynomial a(n) + a(n + m) + a(n + 2*m) + ... + a(n + r*m) in n has its zeros on the vertical line Re(n) = (4 - r*m)/2 in the complex plane. - Peter Bala, Jun 02 2024

			G.f. = x^5 + 6*x^6 + 21*x^7 + 56*x^8 + 126*x^9 + 252*x^10 + 462*x^11 + ...
For A={1,2,3,4}, the only subset with 4 elements is {1,2,3,4}; sum of 2 minimum elements of this subset: a(4) = 1+2 = 3 = 3*C(4+1,5).
For A={1,2,3,4,5}, the subsets with 4 elements are {1,2,3,4}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}; sum of 2 smallest elements of each subset: a(5) = (1+2)+(1+2)+(1+2)+(1+3)+(2+3) = 18 = 3*C(5+1,5). - _Serhat Bulut_, Mar 11 2015
a(6) = 6 from the six independent components of an antisymmetric tensor A of rank 5 and dimension 6: A(1,2,3,4,5), A(1,2,3,4,6), A(1,2,3,5,6), A(1,2,4,5,6), A(1,3,4,5,6), A(2,3,4,5,6). See the Dec 10 2015 comment. - _Wolfdieter Lang_, Dec 10 2015

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).

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].
Serhat Bulut, Subset Sum Problem, 2015.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
C. E. Frasser and G. N. Vostrov, Geodetic Graphs Homeomorphic to a Given Geodetic Graph, arXiv:1611.01873 [cs.DM], 2016. [p. 27]
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 255
H. K. Kim, On Regular Polytope Numbers, Proc. Amer.Math. Soc. 131 (2003), 65-75.
Iva Kodrnja and Helena Koncul, Polynomials vanishing on a basis of S_m(Gamma_0(N)), Glasnik Matematički (2024) Vol. 59, No. 79, 313-325. See p. 324.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901. - _Juergen Will_, Jan 02 2016
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
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; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Composition.
A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A002299, A053127, A000332, A000579, A000580, A000581, A000582.
Cf. A000217, A005583, A051747, A000292.
Cf. A099242. - Gary W. Adamson, Feb 02 2009
Cf. A242023. A104712 (fourth column, k=5).
Cf. A001477, A049310, A052787, A067764, A110555, A277935.
5-cell colorings: A337895 (oriented), A132366(n-1) (achiral).
Unoriented colorings: A063843 (5-cell edges, faces), A128767 (8-cell vertices, 16-cell facets), A337957 (16-cell vertices, 8-cell facets), A338949 (24-cell), A338965 (600-cell vertices, 120-cell facets).
Chiral colorings: A331352 (5-cell edges, faces), A337954 (8-cell vertices, 16-cell facets), A234249 (16-cell vertices, 8-cell facets), A338950 (24-cell), A338966 (600-cell vertices, 120-cell facets).

a000389 n = a000389_list !! n
a000389_list = 0 : 0 : f [] a000217_list where
   f xs (t:ts) = (sum $ zipWith (*) xs a000217_list) : f (t:xs) ts
-- Reinhard Zumkeller, Mar 03 2015, Apr 13 2012

[Binomial(n, 5): n in [0..40]]; // Vincenzo Librandi, Mar 12 2015

f:=n->(1/120)*(n^5-10*n^4+35*n^3-50*n^2+24*n): seq(f(n), n=0..60);
ZL := [S, {S=Prod(B,B,B,B,B,B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n+1), n=0..42); # Zerinvary Lajos, Mar 13 2007
A000389:=1/(z-1)**6; # Simon Plouffe, 1992 dissertation

Table[Binomial[n, 5], {n, 5, 50}] (* Stefan Steinerberger, Apr 02 2006 *)
CoefficientList[Series[x^5 / (1 - x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 12 2015 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,0,0,0,0,1},50] (* Harvey P. Dale, Jul 17 2016 *)

(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)

G.f.: x^5/(1-x)^6.
a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)/120.
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 with 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) = -A110555(n+1,5). - Reinhard Zumkeller, Jul 27 2005
a(n+3) = (1/2!)*(d^2/dx^2)S(n,x)|A049310.%20-%20_Wolfdieter%20Lang">{x=2}, n>=2, one half of second derivative of Chebyshev S-polynomials evaluated at x=2. See A049310. - _Wolfdieter Lang, Apr 04 2007
a(n) = A052787(n+5)/120. - Zerinvary Lajos, Apr 26 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
Sum_{n>=5} (-1)^(n + 1)/a(n) = 80*log(2) - 655/12 = 0.8684411114... - Richard R. Forberg, Aug 11 2014
a(n) = -a(4-n) for all n in Z. - Michael Somos, Oct 07 2014
0 = a(n)*(+a(n+1) + 4*a(n+2)) + a(n+1)*(-6*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Oct 07 2014
a(n) = 3*C(n+1, 5) = 3*A000389(n+1). - Serhat Bulut, Mar 11 2015
From Ilya Gutkovskiy, Jul 23 2016: (Start)
E.g.f.: x^5*exp(x)/120.
Inverse binomial transform of A054849. (End)
From Robert A. Russell, Dec 24 2020: (Start)
a(n) = A337895(n) - a(n+4) = (A337895(n) - A132366(n-1)) / 2 = a(n+4) - A132366(n-1).
a(n+4) = A337895(n) - a(n) = (A337895(n) + A132366(n-1)) / 2 = a(n) + A132366(n-1).
a(n+4) = 1*C(n,1) + 4*C(n,2) + 6*C(n,3) + 4*C(n,4) + 1*C(n,5), where the coefficient of C(n,k) is the number of unoriented pentachoron colorings using exactly k colors. (End)

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

A005583 Coefficients of Chebyshev polynomials.

Original entry on oeis.org

2, 11, 36, 91, 196, 378, 672, 1122, 1782, 2717, 4004, 5733, 8008, 10948, 14688, 19380, 25194, 32319, 40964, 51359, 63756, 78430, 95680, 115830, 139230, 166257, 197316, 232841, 273296, 319176, 371008, 429352, 494802, 567987, 649572, 740259, 840788

Offset: 1

N. J. A. Sloane

If X is an n-set and Y a fixed 2-subset of X then a(n-5) is equal to the number of (n-5)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007

a(n-1) = risefac(n,5)/5! - risefac(n,3)/3! is for n >= 1 also the number of independent components of a symmetric traceless tensor of rank 5 and dimension n. Here risefac is the rising factorial. Put a(0) = 0. - Wolfdieter Lang, Dec 10 2015

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 22.7, p. 797.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..172
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].
Richard K. Guy, Letter to N. J. A. Sloane, Feb 1988.
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.
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [broken link]
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Cached copy, May 15 2013]
Index entries for sequences related to Chebyshev polynomials.

Cf. A000096, A000217, A000389, A051747, A127672.

Column 3 of A207606.

A005583:=-(-2+z)/(z-1)**6; # Simon Plouffe in his 1992 dissertation (this g.f. assumes offset 0)

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));
v=vector(10,i,t(i)-1);
conv(u,v)

a(n) = (1/120)*n*(n+9)*(n+3)*(n+2)*(n+1); \\ Joerg Arndt, Mar 05 2018

G.f.: x*(2-x)/(1-x)^6.
a(n) = binomial(n+4, n-1) + binomial(n+3, n-1) = (1/120)*n*(n+9)*(n+3)*(n+2)*(n+1).
a(n+1) = -A127672(10+n, n), n >= 0, with the coefficients of the Chebyshev C-polynomials A127672(n, k). - Wolfdieter Lang, Dec 10 2015
a(n) = Sum_{i=1..n} A000217(i)*A000096(n+1-i). - Bruno Berselli, Mar 05 2018
a(n) = binomial(n+3,5) + 2*binomial(n+3,4). - Yuchun Ji, May 23 2019
From Amiram Eldar, Feb 17 2023: (Start)
Sum_{n>=1} 1/a(n) = 40751/63504.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1360*log(2)/63 - 922961/63504. (End)

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 07 1999

More terms from Zerinvary Lajos, Jul 21 2006

A102756 Triangle T(n,k), 0<=k<=n, read by rows defined by: T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 10, 10, 3, 5, 20, 31, 20, 5, 6, 35, 76, 78, 40, 8, 7, 56, 161, 232, 184, 76, 13, 8, 84, 308, 582, 636, 406, 142, 21, 9, 120, 546, 1296, 1831, 1604, 861, 260, 34, 10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55

Offset: 0

Philippe Deléham, Dec 18 2006

Rising and falling diagonals are A008999, A124400.
Subtriangle of triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 17 2012
Jointly generated with A209130 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x).  See the Mathematica section. - Clark Kimberling, Mar 05 2012

			Triangle begins:
  1;
  2, 1;
  3, 4, 2;
  4, 10, 10, 3;
  5, 20, 31, 20, 5;
  6, 35, 76, 78, 40, 8;
  7, 56, 161, 232, 184, 76, 13;
  8, 84, 308, 582, 636, 406, 142, 21;
  9, 120, 546, 1296, 1831, 1604, 861, 260, 34;
  10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55;
Triangle (1, 1, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
  1
  1, 0
  2, 1, 0
  3, 4, 2, 0
  4, 10, 10, 3, 0
  5, 20, 31, 20, 5, 0
  6, 35, 76, 78, 40, 8, 0
  7, 56, 161, 232, 184, 76, 13, 0

Cf. A209130.
Cf. A008999, A124400, A084938, A209130.
Cf. A254006, A000012, A000027, A000244, A015530, A015544.
Cf. A001629, A000045, A000292, A051747.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A102756 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209130 *)
(* Clark Kimberling, Mar 05 2012 *)

Sum_{k=0..n} x^k*T(n,k) = A254006(n), A000012(n), A000027(n+1), A000244(n), A015530(n+1), A015544(n+1) for x = -2, -1, 0, 1, 2, 3 respectively.
T(n,n-1) = 2*A001629(n+1) for n>=1.
T(n,n) = Fibonacci(n+1) = A000045(n+1).
T(n,0) = n+1.
T(n,1) = A000292(n) for n>=1.
T(n+1,2) = binomial(n+4,n-1)+binomial(n+2,n-1)= A051747(n) for n>=1.
G.f.: 1/(1-(2+y)*x+(1+y)*(1-y)*x^2). - Philippe Deléham, Feb 17 2012

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A000389 Binomial coefficients C(n,5).

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A005583 Coefficients of Chebyshev polynomials.

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A102756 Triangle T(n,k), 0<=k<=n, read by rows defined by: T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k.

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