A000459 -id:A000459 - cp's OEIS frontend

A059072 Duplicate of A000459.

1, 0, 1, 10, 297, 13756, 925705, 85394646, 10351036465, 1596005408152

Offset: 0

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

dead

A027468 9 times the triangular numbers A000217.

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

Offset: 0

Olivier Gérard

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
Sum of the numbers from 4*n to 5*n. - Wesley Ivan Hurt, Nov 01 2014

			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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Augustine O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., Vol. 308, No. 12 (2008), pp. 2492-2501.
Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.
Leo Tavares, Illustration: Centroid Triangles.
D. Zvonkine, Counting ramified coverings and intersection theory on Hurwitz spaces II (local structure of Hurwitz spaces and combinatorial results), Moscow Mathematical Journal, Vol. 7, No. 1 (2007), pp. 135-162.
D. Zvonkine, Home Page.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for two-way infinite sequences.

Third diagonal of A027465.
Cf. A000459, A002378, A008585, A024966, A028895, A028896, A038764, A033996, A045943, A046092, A049598, A059073, A080855, A134171, A283394.
Cf. A060544.

[9*n*(n+1)/2: n in [0..50]]; // Vincenzo Librandi, Dec 29 2012

[seq(9*binomial(n+1,2), n=0..50)]; # Zerinvary Lajos, Nov 24 2006

Table[(9/2)*n*(n+1), {n,0,50}] (* G. C. Greubel, Aug 22 2017 *)

PARI
```
a(n)=9*n*(n+1)/2
```

[9*binomial(n+1, 2) for n in (0..50)] # G. C. Greubel, May 20 2021

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).
a(n) = 9*C(n+1,2), n>=0. - Zerinvary Lajos, Aug 06 2008
a(n) = a(n-1) + 9*n (with a(0)=0). - Vincenzo Librandi, Nov 19 2010
a(n) = A060544(n+1) - 1. - Omar E. Pol, Oct 03 2011
a(n) = A218470(9*n+8). - Philippe Deléham, Mar 27 2013
E.g.f.: (9/2)*x*(x+2)*exp(x). - G. C. Greubel, Aug 22 2017
a(n) = A060544(n+1) - 1. See Centroid Triangles illustration. - Leo Tavares, Dec 27 2021
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 2/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/9 - 2/9. (End)
From Amiram Eldar, Feb 21 2023: (Start)
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)

More terms from Patrick De Geest, Oct 15 1999

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

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

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

			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.

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.

Michael De Vlieger, Table of n, a(n) for n = 0..100
Shalosh B. Ekhad, Christoph Koutschan, and Doron Zeilberger, There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts], arXiv:2101.10147 [math.CO], 2021.
F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197.
Barbara H. Margolius, Dinner-Diner Matching Probabilities
Barbara H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118.
R. D. McKelvey and A. McLennan, The maximal number of regular totally mixed Nash equilibria, J. Economic Theory, 72 (1997), 411-425.
S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617-620.
Raimundas Vidunas, MacMahon's master theorem and totally mixed Nash equilibria, arXiv preprint arXiv:1401.5400 [math.CO], 2014.
Raimundas Vidunas, Counting derangements and Nash equilibria, Ann. Comb. 21, No. 1, 131-152 (2017).
Index entries for sequences related to card matching

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

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);

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 *)

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.

A000316 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.

Original entry on oeis.org

1, 0, 4, 80, 4752, 440192, 59245120, 10930514688, 2649865335040, 817154768973824, 312426715251262464, 145060238642780180480, 80403174342119992692736, 52443098500204184915312640, 39764049487996490505336537088

Offset: 0

N. J. A. Sloane

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

			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.

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).

T. D. Noe, Table of n, a(n) for n = 0..100
F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197.
B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118.
Barbara H. Margolius, Dinner-Diner Matching Probabilities
L. I. Nicolaescu, Derangements and asymptotics of the Laplace transforms of large powers of a polynomial, New York J. Math. 10 (2004) 117-131.
John Riordan and N. J. A. Sloane, Correspondence, 1974
E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
Index entries for sequences related to card matching

Cf. A000166, A000459, A337302.

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);

(* b = A000459 *)
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;
Array[a, 14] (* Jean-François Alcover, Oct 30 2019 *)

a(n)=if(n==0, 1, round(2^(n/2+3/4)/Pi^.5*exp(-2)*n!*besselk(1/2+n,2^.5))<M. F. Hasler, Sep 27 2015

\\ 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))
a(n) = count(2*n) \\ M. F. Hasler, Sep 30 2015

a(n) = A000459(n)*2^n.
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.
From Dan Dima, Dec 17 2006: (Start)
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
From Peter Bala, Jul 07 2014: (Start)
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)

Formulae, more terms etc. from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 22 2000
Edited by M. F. Hasler, Sep 27 2015 and N. J. A. Sloane, Oct 02 2015
a(0)=1 prepended by Andrew Howroyd, Oct 09 2020

A374980 Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed pair (j,j).

Original entry on oeis.org

1, 0, 5, 74, 2193, 101644, 6840085, 630985830, 76484389121, 11792973495032, 2254432154097861, 523368281765512930, 145044815855963403985, 47302856057098946329284, 17933275902554972391519893, 7820842217155394547769452734, 3887745712142302082441578104705

Offset: 0

Alois P. Heinz, Aug 05 2024

nonn

Inverse binomial transform of A000680.

			a(2) = 5: 1212, 1221, 2112, 2121, 2211.

Alois P. Heinz, Table of n, a(n) for n = 0..238

Cf. A000166, A000459, A000680, A059841, A116218, A375222, A375223.

Column k=2 of A375694.

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);

a(n) = Sum_{j=0..n} (-1)^j*binomial(n,j)*A000680(n-j).
a(n) = A116218(n)/2^n.
a(n) mod 2 = 1 - (n mod 2) = A059841(n).

A372307 Square array read by antidiagonals: T(n,k) is the number of derangements of a multiset comprising n repeats of a k-element set.

Original entry on oeis.org

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

Offset: 0

Jeremy Tan, Apr 26 2024

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.

			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, ...

Jeremy Tan, Antidiagonals n = 0..32, flattened
Shalosh B. Ekhad, Christoph Koutschan and Doron Zeilberger, There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts], arXiv:2101.10147 [math.CO], 2021.
S. Even and J. Gillis, Derangements and Laguerre polynomials, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 79, Issue 1, January 1976, pp. 135-143.
B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), pp. 107-118.
Jeremy Tan, Python program
Raimundas Vidunas, MacMahon's master theorem and totally mixed Nash equilibria, arXiv:1401.5400 [math.CO], 2014.
Raimundas Vidunas, Counting derangements and Nash equilibria, Ann. Comb. 21, No. 1, 131-152 (2017).
Index entries for sequences related to card matching

Rows 0-5 give A000012, A000166, A000459, A059073, A059074, A123297.
Columns 0-4 give A000012, A000007, A000012, A000172, A371252.
Main diagonal gives A375778.

A:= (n, k)-> (-1)^(n*k)*int(exp(-x)*orthopoly[L](n, x)^k, x=0..infinity):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Aug 27 2024

Table[Abs[Integrate[Exp[-x] LaguerreL[n, x]^(s-n), {x, 0, Infinity}]], {s, 0, 9}, {n, 0, s}] // Flatten

Python
```
# See link.
```

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).

T(n,k) ~ A089759(n,k)/exp(n).

A275801 Number of alternating permutations of the multiset {1,1,2,2,...,n,n}.

Original entry on oeis.org

1, 0, 1, 4, 53, 936, 25325, 933980, 45504649, 2824517520, 217690037497, 20394614883316, 2282650939846781, 300814135522967736, 46103574973075123877, 8130996533576437261772, 1635028654501420083152785, 371853339350614571322913824, 94969025880924845123887493233

Offset: 0

Max Alekseyev, Aug 09 2016

nonn

Number of permutations (p(1),...,p(2n)) of {1,1,2,2,...,n,n} satisfying p(1) < p(2) > p(3) < ... < p(2n).

a(n) <= A005799(n) <= A275829(n).

Max Alekseyev, Table of n, a(n) for n = 0..70
Wikipedia, Alternating permutation.
hkju et al., Number of updown sequences of 1,1,2,2,...,n,n, Mathoverflow, 2016.

Column k=2 of A275784.

Cf. A000111, A001250, A000459, A004075, A005799, A114938, A137729, A137730, A137737, A137749, A275829.

A275829 Number of weakly alternating permutations of the multiset {1,1,2,2,...,n,n}.

Original entry on oeis.org

1, 1, 2, 12, 140, 2564, 68728, 2539632, 123686800, 7677924688, 591741636128, 55438330474944, 6204888219697856, 817697605612952384, 125322509904814743424, 22102340129003429880576, 4444468680409243484516608, 1010802175212828388101900544, 258152577318424951261637001728

Offset: 0

Max Alekseyev, Aug 11 2016

nonn

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).

a(n) >= A005799(n) >= A275801(n).

Max Alekseyev, Table of n, a(n) for n = 0..70

Cf. A000111, A001250, A000459, A004075, A005799, A114938, A137729, A137730, A137737, A137749, A275801.

A337303 Number of X-based filling of diagonals in a diagonal Latin square of order n.

Original entry on oeis.org

1, 1, 0, 0, 96, 480, 57600, 403200, 191600640, 1724405760, 1597368729600, 17571056025600, 28378507272192000, 368920594538496000, 952903592436341145600, 14293553886545117184000, 55442575636536644075520000, 942523785821122949283840000, 5231730206388249282710863872000

Offset: 0

Eduard I. Vatutin, Aug 22 2020

nonn

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.

			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

Andrew Howroyd, Table of n, a(n) for n = 0..100
S. Kochemazov, O. Zaikin, E. Vatutin E., and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
E. I. Vatutin, About the number of X-based fillings of diagonals in a diagonal Latin squares of orders 1-15 (in Russian).
E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A000316, A000459, A309283, A274171, A337302.

\\ here b(n) is A000459.
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)! )); }
a(n) = {2^(n\2) * b(n\2) * n!} \\ Andrew Howroyd, Mar 26 2023

a(n) = A337302(n)*n!.

a(n) = n!*A000316(floor(n/2)). - Andrew Howroyd, Mar 26 2023

a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, Mar 26 2023

A124007 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with n-3 fixed points.

Original entry on oeis.org

0, 0, 54, 216, 540, 1080, 1890, 3024, 4536, 6480, 8910, 11880, 15444, 19656, 24570

Offset: 0

Zerinvary Lajos, Nov 01 2006

nonn

			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...

Cf. A059058, A027468, A059073, A000459.

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;

cp's OEIS Frontend

A059072 Duplicate of A000459.

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A027468 9 times the triangular numbers A000217.

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

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A374980 Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed pair (j,j).

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A372307 Square array read by antidiagonals: T(n,k) is the number of derangements of a multiset comprising n repeats of a k-element set.

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A275801 Number of alternating permutations of the multiset {1,1,2,2,...,n,n}.

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