Displaying 1-10 of 16 results found.
1, 0, 1, 10, 297, 13756, 925705, 85394646, 10351036465, 1596005408152
AUTHOR
Barbara Haas Margolius (margolius(AT)math.csuohio.edu)
9 times the triangular numbers A000217.
+10
28
0, 9, 27, 54, 90, 135, 189, 252, 324, 405, 495, 594, 702, 819, 945, 1080, 1224, 1377, 1539, 1710, 1890, 2079, 2277, 2484, 2700, 2925, 3159, 3402, 3654, 3915, 4185, 4464, 4752, 5049, 5355, 5670, 5994, 6327, 6669, 7020, 7380, 7749, 8127, 8514, 8910, 9315
COMMENTS
Staggered diagonal of triangular spiral in A051682, between (0,1,11) spoke and (0,8,25) spoke. - Paul Barry, Mar 15 2003 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with n-2 fixed points. - Zerinvary Lajos, Oct 15 2006 Number of n permutations (n>=2) of 4 objects u, v, z, x with repetition allowed, containing n-2=0 u's. Example: if n=2 then n-2 =zero (0) u, a(1)=9 because we have vv, zz, xx, vx, xv, zx, xz, vz, zv. A027465 formatted as a triangular array: diagonal: 9, 27, 54, 90, 135, 189, 252, 324, ... . - Zerinvary Lajos, Aug 06 2008 a(n) is also the least weight of self-conjugate partitions having n different parts such that each part is a multiple of 3. - Augustine O. Munagi, Dec 18 2008 Also sequence found by reading the line from 0, in the direction 0, 9, ..., and the same line from 0, in the direction 0, 27, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Axis perpendicular to A195147 in the same spiral. - Omar E. Pol, Sep 18 2011
FORMULA
Numerators of sequence a[n, n-2] in (a[i, j])^2 where a[i, j] = binomial(i-1, j-1)/2^(i-1) if j<=i, 0 if j>i.
a(n) = (9/2)*n*(n+1).
a(n) = 9*C(n, 1) + 9*C(n, 2) (binomial transform of (0, 9, 9, 0, 0, ...)). - Paul Barry, Mar 15 2003 G.f.: 9*x/(1-x)^3.
a(-1-n) = a(n).
Sum_{n>=1} 1/a(n) = 2/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/9 - 2/9. (End)
Product_{n>=1} (1 - 1/a(n)) = -(9/(2*Pi))*cos(sqrt(17)*Pi/6).
Product_{n>=1} (1 + 1/a(n)) = 9*sqrt(3)/(4*Pi). (End)
EXAMPLE
The first such self-conjugate partitions, corresponding to a(n)=1,2,3,4 are 3+3+3, 6+6+6+3+3+3, 9+9+9+6+6+6+3+3+3, 12+12+12+9+9+9+6+6+6+3+3+3. - Augustine O. Munagi, Dec 18 2008
MATHEMATICA
Table[(9/2)*n*(n+1), {n, 0, 50}] (* G. C. Greubel, Aug 22 2017 *)
PROG
(PARI) a(n)=9*n*(n+1)/2
(Sage) [9*binomial(n+1, 2) for n in (0..50)] # G. C. Greubel, May 20 2021
CROSSREFS
Cf. A000459, A002378, A008585, A024966, A028895, A028896, A038764, A033996, A045943, A046092, A049598, A059073, A080855, A134171, A283394.
Card-matching numbers (Dinner-Diner matching numbers).
+10
10
1, 0, 1, 56, 13833, 6699824, 5691917785, 7785547001784, 16086070907249329, 47799861987366600992, 196500286135805946117201, 1082973554682091552092493880, 7797122311868240909226166565881
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
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.
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.
EXAMPLE
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.
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 *)
AUTHOR
Barbara Haas Margolius (margolius(AT)math.csuohio.edu)
Two decks each have n kinds of cards, 2 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. a(n) is the number of ways of achieving no matches.
(Formerly M3702 N1513)
+10
9
1, 0, 4, 80, 4752, 440192, 59245120, 10930514688, 2649865335040, 817154768973824, 312426715251262464, 145060238642780180480, 80403174342119992692736, 52443098500204184915312640, 39764049487996490505336537088
COMMENTS
Each deck contains 2n cards.
The probability of no matches is a(n)/(2n)!.
n couples meet for a party and they exchange gifts. Each of the 2n writes their name on a piece of paper and puts it into a hat. They take turns drawing names and give their gift to the person whose name they drew. If anyone draws either their own name or the name of their partner, everyone puts the name they have drawn back into the hat and everyone draws anew. a(n) is the number of different permissible drawings. - Dan Dima, Dec 17 2006 (2n)! / a(n) is the expected number of deck shuffles until no matches occur. a(n) / (2n)! is the probability for a permissible drawing to be achieved. (2n)! / a(n) is the expected number of drawings before a permissible drawing is achieved. As n goes to infinity (2n)! / a(n) will strictly decrease very slowly to e^2 ~ 7.38906 (starting from n > 2) - Dan Dima, Dec 17 2006 a(n) equals the permanent of the (2n)X(2n) matrix with 0's along the main diagonal and the antidiagonal, and 1's everywhere else. - John M. Campbell, Jul 11 2011 Also, number of permutations p of (1,...,2n) such that round(p(k)/2) != round(k/2) for all k=1,...,2n (where half-integers are rounded up). - M. F. Hasler, Sep 30 2015
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, p. 187.
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).
FORMULA
G.f.: Sum_{j=0..n*k} coeff(R(x, n, k), x, j)*(-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 R(x, n, k) is the rook polynomial given by R(x, n, k)=(k!^2*Sum_{j=0..k} x^j/((k-j)!^2*j!))^n (see Riordan). coeff(R(x, n, k), x, j) indicates the coefficient for x^j of the rook polynomial.
a(n) = n! * Sum_{a,b >= 0, a+b <= n} (-1)^b * 2^(a+2*b) * (2*n-2*a-b)! / (a! * b! * (n-a-b)!).
a(n) = n * a(n-1) + n! * 4^n * Sum_{a=0..n} (-1)^a / (a! * 2^a). (End)
a(n) = 2^n * round(2^(n/2 + 3/4)*Pi^(-1/2)*exp(-2)*n!*BesselK(1/2+n,2^(1/2))) for n > 0. - Mark van Hoeij, Oct 30 2011 Recurrence: (2*n-3)*a(n) = 2*(n-1)*(2*n-1)^2*a(n-1) + 4*(n-1)*(2*n-3)*a(n-2) - 16*(n-2)*(n-1)*(2*n-1)*a(n-3). - Vaclav Kotesovec, Aug 07 2013 a(n) = Integral_{x>=0} exp(-x)*(x^2 - 4*x + 2)^n dx. Cf. A000166(n) = Integral_{x>=0} exp(-x)*(x - 1)^n dx. Asymptotic: a(n) ~ (2*n)!*exp(-2)*( 1 - 1/(2*n) - 23/(96*n^2) + O(1/n^3) ). See Nicolaescu. (End)
EXAMPLE
There are 80 ways of achieving zero matches when there are 2 cards of each kind and 3 kinds of card so a(3)=80.
Among the 24 (multiset) permutations of {1,1',2,2'}, only {2,2',1,1'}, {2',2,1,1'}, {2,2',1',1} and {2',2,1',1} have no fixed points, thus a(2)=4.
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), n=0..18);
MATHEMATICA
b[n_] := b[n] = Switch[n, 0, 1, 1, 0, 2, 1, _, n(2n-1) b[n-1] + 2n(n-1) b[n-2] - (2n-1)];
a[n_] := b[n] * 2^n;
PROG
(PARI) a(n)=if(n==0, 1, round(2^(n/2+3/4)/Pi^.5*exp(-2)*n!*besselk(1/2+n, 2^.5))<<n) \\ requires sufficient realprecision. - M. F. Hasler, Sep 27 2015 (PARI) \\ Illustration of the multiset-fixed-point interpretation
count(n, c=ceil(vector(n, i, i)/2))=sum(k=1, n!, !setsearch(Set(ceil(Vec(numtoperm(n, k))/2)-c), 0))
EXTENSIONS
Formulae, more terms etc. from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 22 2000
Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed pair (j,j).
+10
7
1, 0, 5, 74, 2193, 101644, 6840085, 630985830, 76484389121, 11792973495032, 2254432154097861, 523368281765512930, 145044815855963403985, 47302856057098946329284, 17933275902554972391519893, 7820842217155394547769452734, 3887745712142302082441578104705
COMMENTS
Inverse binomial transform of A000680.
FORMULA
a(n) = Sum_{j=0..n} (-1)^j*binomial(n,j)* A000680(n-j). a(n) mod 2 = 1 - (n mod 2) = A059841(n).
EXAMPLE
a(2) = 5: 1212, 1221, 2112, 2121, 2211.
MAPLE
a:= proc(n) option remember; `if`(n<3, [1, 0, 5][n+1],
(n-1)*((2*n+1)*a(n-1)+(4*n-3)*a(n-2)+2*(n-2)*a(n-3)))
end:
seq(a(n), n=0..16);
Square array read by antidiagonals: T(n,k) is the number of derangements of a multiset comprising n repeats of a k-element set.
+10
6
1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 9, 10, 1, 0, 1, 1, 44, 297, 56, 1, 0, 1, 1, 265, 13756, 13833, 346, 1, 0, 1, 1, 1854, 925705, 6699824, 748521, 2252, 1, 0, 1, 1, 14833, 85394646, 5691917785, 3993445276, 44127009, 15184, 1, 0, 1
COMMENTS
A deck has k suits of n cards each. The deck is shuffled and dealt into k hands of n cards each. A match occurs for every card in the i-th hand of suit i. T(n,k) is the number of ways of achieving no matches. The probability of no matches is T(n,k)/((n*k)!/n!^k).
T(n,k) is the maximal number of totally mixed Nash equilibria in games of k players, each with n+1 pure options.
FORMULA
T(n,k) = (-1)^(n*k) * Integral_{x=0..oo} exp(-x)*L_n(x)^k dx, where L_n(x) is the Laguerre polynomial of degree n (Even and Gillis).
EXAMPLE
Square array T(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 0, 1, 2, 9, 44, ...
1, 0, 1, 10, 297, 13756, ...
1, 0, 1, 56, 13833, 6699824, ...
1, 0, 1, 346, 748521, 3993445276, ...
1, 0, 1, 2252, 44127009, 2671644472544, ...
1, 0, 1, 15184, 2750141241, 1926172117389136, ...
1, 0, 1, 104960, 178218782793, 1463447061709156064, ...
MAPLE
A:= (n, k)-> (-1)^(n*k)*int(exp(-x)*orthopoly[L](n, x)^k, x=0..infinity):
MATHEMATICA
Table[Abs[Integrate[Exp[-x] LaguerreL[n, x]^(s-n), {x, 0, Infinity}]], {s, 0, 9}, {n, 0, s}] // Flatten
Number of alternating permutations of the multiset {1,1,2,2,...,n,n}.
+10
4
1, 0, 1, 4, 53, 936, 25325, 933980, 45504649, 2824517520, 217690037497, 20394614883316, 2282650939846781, 300814135522967736, 46103574973075123877, 8130996533576437261772, 1635028654501420083152785, 371853339350614571322913824, 94969025880924845123887493233
COMMENTS
Number of permutations (p(1),...,p(2n)) of {1,1,2,2,...,n,n} satisfying p(1) < p(2) > p(3) < ... < p(2n).
CROSSREFS
Cf. A000111, A001250, A000459, A004075, A005799, A114938, A137729, A137730, A137737, A137749, A275829.
Number of weakly alternating permutations of the multiset {1,1,2,2,...,n,n}.
+10
3
1, 1, 2, 12, 140, 2564, 68728, 2539632, 123686800, 7677924688, 591741636128, 55438330474944, 6204888219697856, 817697605612952384, 125322509904814743424, 22102340129003429880576, 4444468680409243484516608, 1010802175212828388101900544, 258152577318424951261637001728
COMMENTS
Number of permutations (p(1),...,p(2n)) of {1,1,2,2,...,n,n} satisfying p(1) <= p(2) >= p(3) <= p(4) >= p(5) <= ... <= p(2n).
CROSSREFS
Cf. A000111, A001250, A000459, A004075, A005799, A114938, A137729, A137730, A137737, A137749, A275801.
Number of X-based filling of diagonals in a diagonal Latin square of order n.
+10
2
1, 1, 0, 0, 96, 480, 57600, 403200, 191600640, 1724405760, 1597368729600, 17571056025600, 28378507272192000, 368920594538496000, 952903592436341145600, 14293553886545117184000, 55442575636536644075520000, 942523785821122949283840000, 5231730206388249282710863872000
COMMENTS
Used for getting strong canonical forms (SCFs) of the diagonal Latin squares and for fast enumerating of the diagonal Latin squares based on equivalence classes.
EXAMPLE
One of the 96 X-based fillings of diagonals of a diagonal Latin square for order n=4:
1 . . 0
. 0 1 .
. 3 2 .
2 . . 3
PROG
b(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)! )); }
EXTENSIONS
a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, Mar 26 2023
Number of permutations of n distinct letters (ABCD...) each of which appears thrice with n-3 fixed points.
+10
1
0, 0, 54, 216, 540, 1080, 1890, 3024, 4536, 6480, 8910, 11880, 15444, 19656, 24570
EXAMPLE
Maple produces the following triangle - the entries in quotes give the sequence:
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, 2729295, 776520, 180100, 33372, 5355, "540", 90, 0, 1
etc...
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;
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