cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 22 results. Next

A046092 4 times triangular numbers: a(n) = 2*n*(n+1).

Original entry on oeis.org

0, 4, 12, 24, 40, 60, 84, 112, 144, 180, 220, 264, 312, 364, 420, 480, 544, 612, 684, 760, 840, 924, 1012, 1104, 1200, 1300, 1404, 1512, 1624, 1740, 1860, 1984, 2112, 2244, 2380, 2520, 2664, 2812, 2964, 3120, 3280, 3444, 3612, 3784, 3960, 4140, 4324
Offset: 0

Views

Author

N. J. A. Sloane and Eric W. Weisstein

Keywords

Comments

Consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; sequence gives Y values. X values are 1, 3, 5, 7, 9, ... (A005408), Z values are A001844.
In the triple (X, Y, Z) we have X^2=Y+Z. Actually, the triple is given by {x, (x^2 -+ 1)/2}, where x runs over the odd numbers (A005408) and x^2 over the odd squares (A016754). - Lekraj Beedassy, Jun 11 2004
a(n) is the number of edges in n X n square grid with all horizontal and vertical segments filled in. - Asher Auel, Jan 12 2000 [Corrected by Felix Huber, Apr 09 2024]
a(n) is the only number satisfying an inequality related to zeta(2) and zeta(3): Sum_{i>a(n)+1} 1/i^2 < Sum_{i>n} 1/i^3 < Sum_{i>a(n)} 1/i^2. - Benoit Cloitre, Nov 02 2001
Number of right triangles made from vertices of a regular n-gon when n is even. - Sen-Peng Eu, Apr 05 2001
Number of ways to change two non-identical letters in the word aabbccdd..., where there are n type of letters. - Zerinvary Lajos, Feb 15 2005
a(n) is the number of (n-1)-dimensional sides of an (n+1)-dimensional hypercube (e.g., squares have 4 corners, cubes have 12 edges, etc.). - Freek van Walderveen (freek_is(AT)vanwal.nl), Nov 11 2005
From Nikolaos Diamantis (nikos7am(AT)yahoo.com), May 23 2006: (Start)
Consider a triangle, a pentagon, a heptagon, ..., a k-gon where k is odd. We label a triangle with n=1, a pentagon with n=2, ..., a k-gon with n = floor(k/2). Imagine a player standing at each vertex of the k-gon.
Initially there are 2 frisbees, one held by each of two neighboring players. Every time they throw the frisbee to one of their two nearest neighbors with equal probability. Then a(n) gives the average number of steps needed so that the frisbees meet.
I verified this by simulating the processes with a computer program. For example, a(2) = 12 because in a pentagon that's the expected number of trials we need to perform. That is an exercise in Concrete Mathematics and it can be done using generating functions. (End)
A diagonal of A059056. - Zerinvary Lajos, Jun 18 2007
If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n-1) is equal to the number of 2-subsets of X containing none of X_i, (i=1,...,n). - Milan Janjic, Jul 16 2007
X values of solutions to the equation 2*X^3 + X^2 = Y^2. To find Y values: b(n) = 2n(n+1)(2n+1). - Mohamed Bouhamida, Nov 06 2007
Number of (n+1)-permutations of 3 objects u,v,w, with repetition allowed, containing n-1 u's. Example: a(1)=4 because we have vv, vw, wv and ww; a(2)=12 because we can place u in each of the previous four 2-permutations either in front, or in the middle, or at the end. - Zerinvary Lajos, Dec 27 2007
Sequence found by reading the line from 0, in the direction 0, 4, ... and the same line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, May 03 2008
a(n) is also the least weight of self-conjugate partitions having n different even parts. - Augustine O. Munagi, Dec 18 2008
From Peter Luschny, Jul 12 2009: (Start)
The general formula for alternating sums of powers of even integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,1)-(-1)^k P(n,2k+1))/2. Here n=2, thus
a(k) = |(P(2,1) - (-1)^k*P(2,2k+1))/2|. (End)
The sum of squares of n+1 consecutive numbers between a(n)-n and a(n) inclusive equals the sum of squares of n consecutive numbers following a(n). For example, for n = 2, a(2) = 12, and the corresponding equation is 10^2 + 11^2 + 12^2 = 13^2 + 14^2. - Tanya Khovanova, Jul 20 2009
Number of roots in the root system of type D_{n+1} (for n>2). - Tom Edgar, Nov 05 2013
Draw n ellipses in the plane (n>0), any 2 meeting in 4 points; sequence gives number of intersections of these ellipses (cf. A051890, A001844); a(n) = A051890(n+1) - 2 = A001844(n) - 1. - Jaroslav Krizek, Dec 27 2013
a(n) appears also as the second member of the quartet [p0(n), a(n), p2(n), p3(n)] of the square of [n, n+1, n+2, n+3] in the Clifford algebra Cl_2 for n >= 0. p0(n) = -A147973(n+3), p2(n) = A054000(n+1) and p3(n) = A139570(n). See a comment on A147973, also with a reference. - Wolfdieter Lang, Oct 15 2014
a(n) appears also as the third and fourth member of the quartet [p0(n), p0(n), a(n), a(n)] of the square of [n, n, n+1, n+1] in the Clifford algebra Cl_2 for n >= 0. p0(n) = A001105(n). - Wolfdieter Lang, Oct 16 2014
Consider two equal rectangles composed of unit squares. Then surround the 1st rectangle with 1-unit-wide layers to build larger rectangles, and surround the 2nd rectangle just to hide the previous layers. If r(n) and h(n) are the number of unit squares needed for n layers in the 1st case and the 2nd case, then for all rectangles, we have a(n) = r(n) - h(n) for n>=1. - Michel Marcus, Sep 28 2015
When greater than 4, a(n) is the perimeter of a Pythagorean triangle with an even short leg 2*n. - Agola Kisira Odero, Apr 26 2016
Also the number of minimum connected dominating sets in the (n+1)-cocktail party graph. - Eric W. Weisstein, Jun 29 2017
a(n+1) is the harmonic mean of A000384(n+2) and A014105(n+1). - Bob Andriesse, Apr 27 2019
Consider a circular cake from which wedges of equal center angle c are cut out in clockwise succession and turned around so that the bottom comes to the top. This goes on until the cake shows its initial surface again. An interesting case occurs if 360°/c is not an integer. Then, with n = floor(360°/c), the number of wedges which have to be cut out and turned equals a(n). (For the number of cutting line segments see A005408.) - According to Peter Winkler's book "Mathematical Mind-Benders", which presents the problem and its solution (see Winkler, pp. 111, 115) the problem seems to be of French origin but little is known about its history. - Manfred Boergens, Apr 05 2022
a(n-3) is the maximum irregularity over all maximal 2-degenerate graphs with n vertices. The extremal graphs are 2-stars (K_2 joined to n-2 independent vertices). (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, May 29 2023
Number of ways of placing a domino on a (n+1)X(n+1) board of squares. - R. J. Mathar, Apr 24 2024
The sequence terms are the exponents in the expansion of (1/(1 + x)) * Sum_{n >= 0} x^n * Product_{k = 1..n} (1 - x^(2*k-1))/(1 + x^(2*k+1)) = 1 - x^4 + x^12 - x^24 + x^40 - x^60 + - ... (Andrews and Berndt, Entry 9.3.3, p. 229). Cf. A153140. - Peter Bala, Feb 15 2025
Number of edges in an (n+1)-dimensional orthoplex. 2D orthoplexes (diamonds) have 4 edges, 3D orthoplexes (octahedrons) have 12 edges, 4D orthoplexes (16-cell) have 24 edges, and so on. - Aaron Franke, Mar 23 2025

Examples

			a(7)=112 because 112 = 2*7*(7+1).
The first few triples are (1,0,1), (3,4,5), (5,12,13), (7,24,25), ...
The first such partitions, corresponding to a(n)=1,2,3,4, are 2+2, 4+4+2+2, 6+6+4+4+2+2, 8+8+6+6+4+4+2+2. - _Augustine O. Munagi_, Dec 18 2008

References

  • George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, 2005.
  • Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
  • Albert H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 125, 1964.
  • Ronald L. Graham, D. E. Knuth and Oren Patashnik, Concrete Mathematics, Reading, Massachusetts: Addison-Wesley, 1994.
  • Peter Winkler, Mathematical Mind-Benders, Wellesley, Massachusetts: A K Peters, 2007.

Links

Crossrefs

Cf. A045943, A028895, A002943, A054000, A000330, A007290, A002378, A033996, A124080, A028896, A049598, A005563, A000217, A033586, A085250.
Main diagonal of array in A001477.
Equals A033996/2. Cf. A001844. - Augustine O. Munagi, Dec 18 2008
Cf. A078371, A141530 (see Librandi's comment in A078371).
Cf. A097080, A001845.
Cf. similar sequences listed in A299645.
Cf. A005408.
Cf. A016754.
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

Programs

Formula

a(n) = A100345(n+1, n-1) for n>0.
a(n) = 2*A002378(n) = 4*A000217(n). - Lekraj Beedassy, May 25 2004
a(n) = C(2n, 2) - n = 4*C(n, 2). - Zerinvary Lajos, Feb 15 2005
From Lekraj Beedassy, Jun 04 2006: (Start)
a(n) - a(n-1)=4*n.
Let k=a(n). Then a(n+1) = k + 2*(1 + sqrt(2k + 1)). (End)
Array read by rows: row n gives A033586(n), A085250(n+1). - Omar E. Pol, May 03 2008
O.g.f.:4*x/(1-x)^3; e.g.f.: exp(x)*(2*x^2+4*x). - Geoffrey Critzer, May 17 2009
From Stephen Crowley, Jul 26 2009: (Start)
a(n) = 1/int(-(x*n+x-1)*(step((-1+x*n)/n)-1)*n*step((x*n+x-1)/(n+1)),x=0..1) where step(x)=piecewise(x<0,0,0<=x,1) is the Heaviside step function.
Sum_{n>=1} 1/a(n) = 1/2. (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=4, a(2)=12. - Harvey P. Dale, Jul 25 2011
For n > 0, a(n) = 1/(Integral_{x=0..Pi/2} (sin(x))^(2*n-1)*(cos(x))^3). - Francesco Daddi, Aug 02 2011
a(n) = A001844(n) - 1. - Omar E. Pol, Oct 03 2011
(a(n) - A000217(k))^2 = A000217(2n-k)*A000217(2n+1+k) - (A002378(n) - A000217(k)), for all k. See also A001105. - Charlie Marion, May 09 2013
From Ivan N. Ianakiev, Aug 30 2013: (Start)
a(n)*(2m+1)^2 + a(m) = a(n*(2m+1)+m), for any nonnegative integers n and m.
t(k)*a(n) + t(k-1)*a(n+1) = a((n+1)*(t(k)-t(k-1)-1)), where k>=2, n>=1, t(k)=A000217(k). (End)
a(n) = A245300(n,n). - Reinhard Zumkeller, Jul 17 2014
2*a(n)+1 = A016754(n) = A005408(n)^2, the odd squares. - M. F. Hasler, Oct 02 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) - 1/2 = A187832. - Ilya Gutkovskiy, Mar 16 2017
a(n) = lcm(2*n,2*n+2). - Enrique Navarrete, Aug 30 2017
a(n)*a(n+k) + k^2 = m^2 (a perfect square), n >= 1, k >= 0. - Ezhilarasu Velayutham, May 13 2019
From Amiram Eldar, Jan 29 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = cosh(Pi/2)/(Pi/2).
Product_{n>=1} (1 - 1/a(n)) = -2*cos(sqrt(3)*Pi/2)/Pi. (End)
a(n) = A016754(n) - A001844(n). - Leo Tavares, Sep 20 2022

A059071 Card-matching numbers (Dinner-Diner matching numbers) for 5 kinds of cards.

Original entry on oeis.org

1, 44, 45, 20, 10, 0, 1, 440192, 975360, 1035680, 696320, 329600, 114176, 31040, 5120, 1280, 0, 32, 52097831424, 179811290880, 298276007040, 315423836640, 237742646400, 135296008128, 60059024640
Offset: 0

Views

Author

Barbara Haas Margolius (margolius(AT)math.csuohio.edu)

Keywords

Comments

This is a triangle of card matching numbers. Two decks each have 5 kinds of cards, n of each kind. The first deck is laid out in order. The second deck is shuffled and laid out next to the first. A match occurs if a card from the second deck is next to a card of the same kind from the first deck. Triangle T(n,k) is the number of ways of achieving exactly k matches (k=0..5n). The probability of exactly k matches is T(n,k)/(5n)!.
Rows are of length 1,6,11,16,... = 5n+1 = A016861(n). - M. F. Hasler, Sep 20 2015

Examples

			There are 1,035,680 ways of matching exactly 2 cards when there are 2 cards of each kind and 5 kinds of card so T(2,2)=1,035,680.

References

  • F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
  • R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.

Links

Crossrefs

Cf. A008290, A059056-A059071.

Programs

  • Maple
    p := (x,k)->k!^2*sum(x^j/((k-j)!^2*j!),j=0..k); R := (x,n,k)->p(x,k)^n; f := (t,n,k)->sum(coeff(R(x,n,k),x,j)*(t-1)^j*(n*k-j)!,j=0..n*k);
for n from 0 to 3 do seq(coeff(f(t,5,n),t,m),m=0..5*n); od;
  • Mathematica
    p[x_, k_] := k!^2*Sum[ x^j/((k-j)!^2*j!), {j, 0, k}]; r[x_, n_, k_] := p[x, k]^n; f[t_, n_, k_] := Sum[ Coefficient[ r[x, n, k], x, j]*(t-1)^j*(n*k-j)!, {j, 0, n*k}]; Table[ Coefficient[ f[t, 5, n], t, m], {n, 0, 3}, {m, 0, 5*n}] // Flatten (* Jean-François Alcover, Mar 04 2013, translated from Maple *)

Formula

G.f.: sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k) where n is the number of kinds of cards (5 in this case), k is the number of cards of each kind and R(x, n, k) is the rook polynomial given by R(x, n, k)=(k!^2*sum(x^j/((k-j)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the coefficient for x^j of the rook polynomial.

Extensions

Edited by M. F. Hasler, Sep 20 2015

A000459 Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed points.

Original entry on oeis.org

1, 0, 1, 10, 297, 13756, 925705, 85394646, 10351036465, 1596005408152, 305104214112561, 70830194649795010, 19629681235869138841, 6401745422388206166420, 2427004973632598297444857, 1058435896607583305978409166, 526149167104704966948064477665
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Original definition: Number of permutations with no hits on 2 main diagonals. (Identical to definition of A000316.) - M. F. Hasler, Sep 27 2015
Card-matching numbers (Dinner-Diner matching numbers): A deck has n kinds of cards, 2 of each kind. The deck is shuffled and dealt in to n hands with 2 cards each. A match occurs for every card in the j-th hand of kind j. A(n) is the number of ways of achieving no matches. The probability of no matches is A(n)/((2n)!/2!^n).
Also, Penrice's Christmas gift numbers (see Penrice 1991).
a(n) is the maximal number of totally mixed Nash equilibria in games of n players, each with 3 pure options. - Raimundas Vidunas, Jan 22 2014

Examples

			There are 297 ways of achieving zero matches when there are 2 cards of each kind and 4 kinds of card so a(4)=297.
From _Peter Bala_, Jul 08 2014: (Start)
a(3) = 10: the 10 permutations of the multiset {1,1,2,2,3,3} that have no fixed points are
{2,2,3,3,1,1}, {3,3,1,1,2,2}
{2,3,1,3,1,2}, {2,3,1,3,2,1}
{2,3,3,1,1,2}, {2,3,3,1,2,1}
{3,2,1,3,1,2}, {3,2,1,3,2,1}
{3,2,3,1,1,2}, {3,2,3,1,2,1}
(End)

References

  • F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
  • R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.
  • 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

Crossrefs

Cf. A000166, A008290, A059056-A059071, A033581, A059073.
Row n=2 of A372307.

Programs

  • Magma
    I:=[0,1]; [n le 2 select I[n] else n*(2*n-1)*Self(n-1)+2*n*(n-1)*Self(n-2)-(2*n-1): n in [1..30]]; // Vincenzo Librandi, Sep 28 2015
  • Maple
    p := (x,k)->k!^2*sum(x^j/((k-j)!^2*j!),j=0..k); R := (x,n,k)->p(x,k)^n; f := (t,n,k)->sum(coeff(R(x,n,k),x,j)*(t-1)^j*(n*k-j)!,j=0..n*k); seq(f(0,n,2)/2!^n,n=0..18);
  • Mathematica
    RecurrenceTable[{(2*n+3)*a[n+3]==(2*n+5)^2*(n+2)*a[n+2]+(2*n+3)*(n+2)*a[n+1]-2*(2*n+5)*(n+1)*(n+2)*a[n],a[1]==0,a[2]==1,a[3]==10},a,{n,1,25}] (* Vaclav Kotesovec, Aug 31 2012 *)
a[n_] := a[n] = n*(2*n-1)*a[n-1] + 2*n*(n-1)*a[n-2] - (2*n-1); a[0] = 1; a[1] = 0; a[2] = 1; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Mar 04 2013 *)
a[n_] := Sum[(2*(n-m))! / 2^(n-m) Binomial[n, m] Hypergeometric1F1[m-n, 2*(m - n), -4], {m, 0, n}]; Table[a[n], {n, 0, 16}] (* Peter Luschny, Nov 15 2023 *)
  • PARI
    a(n) = (2^n*round(2^(n/2+3/4)*Pi^(-1/2)*exp(-2)*n!*besselk(1/2+n,2^(1/2))))/2^n;
vector(15, n, a(n))\\ Altug Alkan, Sep 28 2015
  • PARI
    { A000459(n) = sum(m=0,n, sum(k=0,n-m, (-1)^k * binomial(n,k) * binomial(n-k,m) * 2^(2*k+m-n) * (2*n-2*m-k)! )); } \\ Max Alekseyev, Oct 06 2016

Formula

a(n) = A000316(n)/2^n.
a(n) = Sum_{k=0..n} Sum_{m=0..n-k} (-1)^k * n!/(k!*m!*(n-k-m)!) * 2^(2*k+m-n) * (2*n-2*m-k)!. - Max Alekseyev, Oct 06 2016
G.f.: Sum_{j=0..n*k} coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)! where n is the number of kinds of cards, k is the number of cards of each kind (2 in this case) and coeff(R(x, n, k), x, j) is the coefficient of x^j of the rook polynomial R(x, n, k) = (k!^2*sum(x^j/((k-j)!^2*j!))^n (see Riordan or Stanley).
D-finite with recurrence a(n) = n*(2*n-1)*a(n-1)+2*n*(n-1)*a(n-2)-(2*n-1), a(1) = 0, a(2) = 1.
a(n) = round(2^(n/2 + 3/4)*Pi^(-1/2)*exp(-2)*n!*BesselK(1/2+n,2^(1/2))). - Mark van Hoeij, Oct 30 2011
(2*n+3)*a(n+3)=(2*n+5)^2*(n+2)*a(n+2)+(2*n+3)*(n+2)*a(n+1)-2*(2*n+5)*(n+1)*(n+2)*a(n). - Vaclav Kotesovec, Aug 31 2012
Asymptotic: a(n) ~ n^(2*n)*2^(n+1)*sqrt(Pi*n)/exp(2*n+2), Vaclav Kotesovec, Aug 31 2012
a(n) = (1/2^n)*A000316(n) = int_{0..inf} exp(-x)*(1/2*x^2 - 2*x + 1)^n dx. Asymptotic: a(n) ~ ((2*n)!/2^n)*exp(-2)*( 1 - 1/(2*n) - 23/(96*n^2) + O(1/n^3) ). See Nicolaescu. - Peter Bala, Jul 07 2014
Let S = x_1 + ... + x_n. a(n) equals the coefficient of (x_1*...*x_n)^2 in the expansion of product {i = 1..n} (S - x_i)^2 (MacMahon, Chapter III). - Peter Bala, Jul 08 2014
Conjecture: a(n+k) - a(n) is divisible by k. - Mark van Hoeij, Nov 15 2023

Extensions

More terms and edited by Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 22 2000
Edited by M. F. Hasler, Sep 27 2015
a(0)=1 prepended by Max Alekseyev, Oct 06 2016

A102860 Number of ways to change three non-identical letters in the word aabbccdd..., where there are n types of letters.

Original entry on oeis.org

0, 16, 64, 160, 320, 560, 896, 1344, 1920, 2640, 3520, 4576, 5824, 7280, 8960, 10880, 13056, 15504, 18240, 21280, 24640, 28336, 32384, 36800, 41600, 46800, 52416, 58464, 64960, 71920, 79360, 87296, 95744, 104720, 114240, 124320, 134976, 146224
Offset: 2

Views

Author

Zerinvary Lajos, Mar 01 2005

Keywords

Comments

There are two ways to change abc: abc -> bca and abc -> cab, that's why we get 2*C(2n,3). There are 2n*(2n-2) = 4n*(n-1) = 8*C(n,2) cases when the two chosen letters are identical, that's why we get -8*C(n,2). Thanks to Miklos Kristof for help.
A diagonal of A059056. - Zerinvary Lajos, Jun 18 2007
With offset "1", a(n) is 16 times the self convolution of n. - Wesley Ivan Hurt, Apr 06 2015
Number of orbits of Aut(Z^7) as function of the infinity norm (n+2) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 53760. - Philippe A.J.G. Chevalier, Dec 28 2015

Examples

			a(4) = 64 = 2*C(8,3) - 8*C(4,2) = 2*56 - 8*6 = 112 - 48.

Links

Crossrefs

Cf. A000292, A046092, A059056.

Programs

Formula

a(n) = 16*C(n, 3) = 2*C(2*n, 3) - 8*C(n, 2).
From R. J. Mathar, Mar 09 2009: (Start)
G.f.: 16*x^3/(1-x)^4.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4).
a(n) = 8*n*(n-1)*(n-2)/3. (End)
a(n) = 16*A000292(n-2). - J. M. Bergot, May 29 2014
E.g.f.: 8*exp(x)*x^3/3. - Stefano Spezia, May 19 2021
From Amiram Eldar, Sep 04 2022: (Start)
Sum_{n>=3} 1/a(n) = 3/32.
Sum_{n>=3} (-1)^(n+1)/a(n) = 3*(8*log(2)-5)/32. (End)

A059073 Card-matching numbers (Dinner-Diner matching numbers).

Original entry on oeis.org

1, 0, 1, 56, 13833, 6699824, 5691917785, 7785547001784, 16086070907249329, 47799861987366600992, 196500286135805946117201, 1082973554682091552092493880, 7797122311868240909226166565881
Offset: 0

Views

Author

Barbara Haas Margolius (margolius(AT)math.csuohio.edu)

Keywords

Comments

A deck has n kinds of cards, 3 of each kind. The deck is shuffled and dealt in to n hands with 3 cards each. A match occurs for every card in the j-th hand of kind j. A(n) is the number of ways of achieving no matches. The probability of no matches is A(n)/((3n)!/3!^n).
Number of fixed-point-free permutations of n distinct letters (ABCD...), each of which appears thrice. If there is only one letter of each type we get A000166. - Zerinvary Lajos, Oct 15 2006
a(n) is the maximal number of totally mixed Nash equilibria in games of n players, each with 4 pure options. - Raimundas Vidunas, Jan 22 2014

Examples

			There are 56 ways of achieving zero matches when there are 3 cards of each kind and 3 kinds of card so a(3)=56.

References

  • F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
  • R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.

Links

Crossrefs

Cf. A000166, A000459, A008290, A059056-A059071.

Programs

  • Maple
    p := (x,k)->k!^2*sum(x^j/((k-j)!^2*j!),j=0..k); R := (x,n,k)->p(x,k)^n; f := (t,n,k)->sum(coeff(R(x,n,k),x,j)*(t-1)^j*(n*k-j)!,j=0..n*k); seq(f(0,n,3)/3!^n,n=0..18);
  • Mathematica
    p[x_, k_] := k!^2*Sum[x^j/((k-j)!^2*j!), {j, 0, k}]; R[x_, n_, k_] := p[x, k]^n; f[t_, n_, k_] := Sum[Coefficient[R[x, n, k], x, j]*(t-1)^j*(n*k-j)!, {j, 0, n*k}]; Table[f[0, n, 3]/3!^n, {n, 0, 12}] (* Jean-François Alcover, May 21 2012, translated from Maple *)

Formula

G.f.: Sum_{j=0..n*k} coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)! where n is the number of kinds of cards, k is the number of cards of each kind (3 in this case) and R(x, n, k) is the rook polynomial given by R(x, n, k) = k!^2*Sum(x^j/((k-j)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the coefficient for x^j of the rook polynomial.

A059058 Card-matching numbers (Dinner-Diner matching numbers).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 9, 0, 9, 0, 1, 56, 216, 378, 435, 324, 189, 54, 27, 0, 1, 13833, 49464, 84510, 90944, 69039, 38448, 16476, 5184, 1431, 216, 54, 0, 1, 6699824, 23123880, 38358540, 40563765, 30573900, 17399178, 7723640
Offset: 0

Views

Author

Barbara Haas Margolius (margolius(AT)math.csuohio.edu)

Keywords

Comments

This is a triangle of card matching numbers. A deck has n kinds of cards, 3 of each kind. The deck is shuffled and dealt in to n hands with 3 cards each. A match occurs for every card in the j-th hand of kind j. Triangle T(n,k) is the number of ways of achieving exactly k matches (k=0..3n). The probability of exactly k matches is T(n,k)/((3n)!/(3!)^n).
Rows have lengths 1,4,7,10,...
Analogous to A008290 - Zerinvary Lajos, Jun 22 2005

Examples

			There are 9 ways of matching exactly 2 cards when there are 2 different kinds of cards, 3 of each in each of the two decks so T(2,2)=9.

References

  • F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
  • R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.

Links

Crossrefs

Cf. A008290, A059056-A059071, A008290.

Programs

  • Maple
    p := (x,k)->k!^2*sum(x^j/((k-j)!^2*j!),j=0..k); R := (x,n,k)->p(x,k)^n; f := (t,n,k)->sum(coeff(R(x,n,k),x,j)*(t-1)^j*(n*k-j)!,j=0..n*k);
for n from 0 to 6 do seq(coeff(f(t,n,3),t,m)/3!^n,m=0..3*n); od;
  • Mathematica
    p[x_, k_] := k!^2*Sum[ x^j/((k-j)!^2*j!), {j, 0, k}]; f[t_, n_, k_] := Sum[ Coefficient[ p[x, k]^n, x, j]*(t-1)^j*(n*k-j)!, {j, 0, n*k}]; Flatten[ Table[ Coefficient[ f[t, n, 3], t, m]/3!^n, {n, 0, 6}, {m, 0, 3n}]] (* Jean-François Alcover, Jan 31 2012, after Maple *)

Formula

G.f.: sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k) where n is the number of kinds of cards, k is the number of cards of each kind (here k is 3) and R(x, n, k) is the rook polynomial given by R(x, n, k)=(k!^2*sum(x^j/((k-j)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the j-th coefficient on x of the rook polynomial.

A000489 Card matching: Coefficients B[n,3] of t^3 in the reduced hit polynomial A[n,n,n](t).

Original entry on oeis.org

1, 16, 435, 7136, 99350, 1234032, 14219212, 155251840, 1628202762, 16550991200, 164111079110, 1594594348800, 15235525651840, 143518352447680, 1335670583147400, 12301278983461376, 112264111607438906, 1016361486936571680, 9136254276320346046
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The definition uses notations of Riordan (1958), except for use of n instead of p. - M. F. Hasler, Sep 22 2015

References

  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 193.
  • 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

Crossrefs

Cf. A000279, A000535.
Cf. A059056 - A059071.
Cf. A001181.

Programs

  • Magma
    [1, 16] cat [&+[3*Binomial(n, 3)*Binomial(n, k+3)*Binomial(n, k)*Binomial(n-3, k) + 6*n*Binomial(n, 2)*Binomial(n, k+1)*Binomial(n-1, k+2)*Binomial(n-2, k): k in [0..n-3]] + &+[n^3*Binomial(n-1, k)^3: k in [0..n-1]]: n in [3..20]]; // Vincenzo Librandi, Sep 22 2015
  • Mathematica
    a[n_] := 3*Binomial[n, 3]*Sum[Binomial[n, k + 3]*Binomial[n, k]*Binomial[n - 3, k], {k, 0, n - 3}] + 6 n*Binomial[n, 2]*Sum[Binomial[n, k + 1]*Binomial[n - 1, k + 2]*Binomial[n - 2, k], {k, 0, n - 3}] + n^3*Sum[Binomial[n - 1, k]^3, {k, 0, n - 1}]; Table[a[n], {n, 20}] (* T. D. Noe, Jun 20 2012 *)
  • PARI
    A000489(n)={3*binomial(n, 3)*sum(k=0,n-3,binomial(n, k+3)*binomial(n, k)*binomial(n-3, k))+6*n*binomial(n, 2)*sum(k=0,n-3,binomial(n, k+1)*binomial(n-1, k+2)*binomial(n-2, k))+n^3*sum(k=0,n-1,binomial(n-1, k)^3)} \\ M. F. Hasler, Sep 20 2015

Formula

a(n) = 3*binomial(n, 3)*sum(binomial(n, k+3)*binomial(n, k)*binomial(n-3, k), k=0..n-3) + 6n*binomial(n, 2)*sum(binomial(n, k+1)*binomial(n-1, k+2)*binomial(n-2, k), k=0..n-3) + n^3*sum(binomial(n-1, k)^3, k=0..n-1).
Recurrence: (n+3)*(243*n^7 - 1701*n^6 + 4239*n^5 - 4671*n^4 + 6042*n^3 - 17352*n^2 + 25032*n - 12016)*(n-1)^2*a(n) = n*(1701*n^9 - 6804*n^8 + 270*n^7 + 19116*n^6 + 35085*n^5 - 203640*n^4 + 324384*n^3 - 246736*n^2 + 75440*n - 5440)*a(n-1) + 8*n*(243*n^7 - 864*n^5 - 486*n^4 + 4233*n^3 - 5274*n^2 + 2460*n - 184)*(n-1)^2*a(n-2). - Vaclav Kotesovec, Aug 07 2013
a(n) ~ 3*sqrt(3)*n^2*8^(n-1)/Pi. - Vaclav Kotesovec, Aug 07 2013
a(n) = n^2*((27*n^3+54*n^2-57*n+8)*(n+2)*A001181(n)-(189*n^3+189*n^2-30*n+16)*(n-1)*A001181(n-1))/96. - Mark van Hoeij, Nov 14 2023

Extensions

More terms from Vladeta Jovovic, Apr 26 2000
More terms from Emeric Deutsch, Feb 19 2004
Definition made more precise by M. F. Hasler, Sep 22 2015

A059060 Card-matching numbers (Dinner-Diner matching numbers).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 0, 16, 0, 36, 0, 16, 0, 1, 346, 1824, 4536, 7136, 7947, 6336, 3936, 1728, 684, 128, 48, 0, 1, 748521, 3662976, 8607744, 12880512, 13731616, 11042688, 6928704, 3458432, 1395126, 453888, 122016, 25344, 4824, 512, 96, 0, 1, 3993445276
Offset: 0

Views

Author

Barbara Haas Margolius (margolius(AT)math.csuohio.edu)

Keywords

Comments

This is a triangle of card matching numbers. A deck has n kinds of cards, 4 of each kind. The deck is shuffled and dealt in to n hands with 4 cards each. A match occurs for every card in the j-th hand of kind j. Triangle T(n,k) is the number of ways of achieving exactly k matches (k=0..4n). The probability of exactly k matches is T(n,k)/((4n)!/(4!)^n).
Rows have lengths 1,5,9,13,...
Analogous to A008290 - Zerinvary Lajos, Jun 22 2005

Examples

			There are 16 ways of matching exactly 2 cards when there are 2 different kinds of cards, 4 of each so T(2,2)=16.
From _Joerg Arndt_, Nov 08 2020: (Start)
The first few rows are
1
0, 0, 0, 0, 1
1, 0, 16, 0, 36, 0, 16, 0, 1
346, 1824, 4536, 7136, 7947, 6336, 3936, 1728, 684, 128, 48, 0, 1
748521, 3662976, 8607744, 12880512, 13731616, 11042688, 6928704, 3458432, 1395126, 453888, 122016, 25344, 4824, 512, 96, 0, 1
3993445276, 18743463360, 42506546320, 61907282240, 64917874125, 52087325696, 33176621920, 17181584640, 7352761180, 2628808000, 790912656, 201062080, 43284010, 7873920, 1216000, 154496, 17640, 1280, 160, 0, 1 (End)

References

  • F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
  • R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.

Links

Crossrefs

Cf. A008290, A059056-A059071.

Programs

  • Maple
    p := (x,k)->k!^2*sum(x^j/((k-j)!^2*j!),j=0..k); R := (x,n,k)->p(x,k)^n; f := (t,n,k)->sum(coeff(R(x,n,k),x,j)*(t-1)^j*(n*k-j)!,j=0..n*k);
for n from 0 to 5 do seq(coeff(f(t,n,4),t,m)/4!^n,m=0..4*n); od;
  • Mathematica
    p[x_, k_] := k!^2*Sum[ x^j/((k-j)!^2*j!), {j, 0, k}]; r[x_, n_, k_] := p[x, k]^n; f[t_, n_, k_] := Sum[ Coefficient[r[x, n, k], x, j]*(t-1)^j*(n*k-j)!, {j, 0, n*k}]; Table[ Coefficient[f[t, n, 4], t, m]/4!^n, {n, 0, 5}, {m, 0, 4*n}] // Flatten (* Jean-François Alcover, Feb 22 2013, translated from Maple *)

Formula

G.f.: sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k) where n is the number of kinds of cards, k is the number of cards of each kind (here k is 4) and R(x, n, k) is the rook polynomial given by R(x, n, k)=(k!^2*sum(x^j/((k-j)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the j-th coefficient on x of the rook polynomial.

A059062 Card-matching numbers (Dinner-Diner matching numbers).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 0, 25, 0, 100, 0, 100, 0, 25, 0, 1, 2252, 15150, 48600, 99350, 144150, 156753, 131000, 87075, 45000, 19300, 6000, 1800, 250, 75, 0, 1, 44127009, 274314600, 822998550, 1583402400, 2189652825, 2311947008
Offset: 0

Views

Author

Barbara Haas Margolius (margolius(AT)math.csuohio.edu)

Keywords

Comments

This is a triangle of card matching numbers. A deck has n kinds of cards, 5 of each kind. The deck is shuffled and dealt in to n hands with 5 cards each. A match occurs for every card in the j-th hand of kind j. Triangle T(n,k) is the number of ways of achieving exactly k matches (k=0..5n). The probability of exactly k matches is T(n,k)/((5n)!/(5!)^n).
Rows have lengths 1,6,11,16,...
Analogous to A008290 - Zerinvary Lajos, Jun 22 2005

Examples

			There are 25 ways of matching exactly 2 cards when there are 2 different kinds of cards, 5 of each so T(2,2)=25.

References

  • F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
  • R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.

Links

Crossrefs

Cf. A008290, A059056-A059071.
Cf. A008290.

Programs

  • Maple
    p := (x,k)->k!^2*sum(x^j/((k-j)!^2*j!),j=0..k); R := (x,n,k)->p(x,k)^n; f := (t,n,k)->sum(coeff(R(x,n,k),x,j)*(t-1)^j*(n*k-j)!,j=0..n*k);
for n from 0 to 4 do seq(coeff(f(t,n,5),t,m)/5!^n,m=0..5*n); od;
  • Mathematica
    nmax = 4; r[x_, n_, k_] := (k!^2*Sum[ x^j/((k-j)!^2*j!), {j, 0, k}])^n; f[t_, n_, k_] := Sum[ Coefficient[ r[x, n, k], x, j]*(t-1)^j*(n*k-j)!, {j, 0, n*k}]; Flatten[ Table[ Coefficient[ f[t, n, 5], t, m]/5!^n, {n, 0, nmax}, {m, 0, 5n}]](* Jean-François Alcover, Nov 23 2011, after Maple *)

Formula

G.f.: sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k) where n is the number of kinds of cards, k is the number of cards of each kind (here k is 5) and R(x, n, k) is the rook polynomial given by R(x, n, k)=(k!^2*sum(x^j/((k-j)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the j-th coefficient on x of the rook polynomial.

A371252 Number of derangements of a multiset comprising n repeats of a 4-element set.

Original entry on oeis.org

1, 9, 297, 13833, 748521, 44127009, 2750141241, 178218782793, 11887871843817, 810822837267729, 56289612791763297, 3964402453931011233, 282558393168537751929, 20342533966643026042641, 1477174422125162468055897, 108064155440237168218117833, 7956914294959071176435002857
Offset: 0

Views

Author

Jeremy Tan, Mar 16 2024

Keywords

Comments

A deck has 4 suits of n cards each. The deck is shuffled and dealt into 4 hands of n cards each. A match occurs for every card in the i-th hand of suit i. a(n) is the number of ways of achieving no matches. The probability of no matches is a(n)/((4n)!/n!^4).

Examples

			There are a(13) = 20342533966643026042641 bridge deals where North, South, East and West are void in clubs, diamonds, hearts and spades, respectively.

Links

Crossrefs

Column k=0 of A059068. The analogous sequence with 3 suits is A000172 and that with 2 suits is A000012.
Column k=4 of A372307.
Cf. A008290, A059056-A059071.

Programs

  • Mathematica
    Table[Integrate[Exp[-x] LaguerreL[n, x]^4, {x, 0, Infinity}], {n, 0, 16}]
(* or *)
rec = n^3(2n-1)(5n-6)(10n-13) a[n] == (8300n^6-37350n^5+66698n^4-60393n^3+29297n^2-7263n+738) a[n-1] - (n-1)(16300n^5-81500n^4+151553n^3-123364n^2+39501n-4338) a[n-2] + 162(n-2)^3(n-1)(5n-1)(10n-3) a[n-3];
RecurrenceTable[{rec, a[0] == 1, a[1] == 9, a[2] == 297}, a, {n, 0, 16}]
  • Python
    def A371252(n):
    l = [1, 9, 297]
    for k in range(3, n+1):
        m1 = (((((8300*k-37350)*k+66698)*k-60393)*k+29297)*k-7263)*k+738
        m2 = (k-1)*(((((16300*k-81500)*k+151553)*k-123364)*k+39501)*k-4338)
        m3 = 162*(k-2)**3*(k-1)*(5*k-1)*(10*k-3)
        r = (m1*l[-1] - m2*l[-2] + m3*l[-3]) // (k**3*(2*k-1)*(5*k-6)*(10*k-13))
        l.append(r)
    return l[n]

Formula

a(n) = Integral_{x=0..oo} exp(-x)*L_n(x)^4 dx, where L_n(x) is the Laguerre polynomial of degree n (Even and Gillis).
D-finite with recurrence n^3*(2*n-1)*(5*n-6)*(10*n-13)*a(n) = (8300*n^6 - 37350*n^5 + 66698*n^4 - 60393*n^3 + 29297*n^2 - 7263*n + 738)*a(n-1) - (n-1)*(16300*n^5 - 81500*n^4 + 151553*n^3 - 123364*n^2 + 39501*n - 4338)*a(n-2) + 162*(n-2)^3*(n-1)*(5*n-1)*(10*n-3)*a(n-3) (Ekhad).
a(n) = [(w*x*y*z)^n] ((x+y+z)*(w+y+z)*(w+x+z)*(w+x+y))^n.
a(n) ~ 3^(4*n + 3) / (32 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Mar 29 2024
Showing 1-10 of 22 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.