A059073 -id:A059073 - cp's OEIS frontend

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

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

N. J. A. Sloane

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

			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)

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).

T. D. Noe, Table of n, a(n) for n = 0..100
Per Alexandersson and Frether Getachew, An involution on derangements, arXiv:2105.08455 [math.CO], 2021.
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.
P. A. MacMahon, Combinatory Analysis Cambridge: The University Press 1915-1916
B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118.
B. H. Margolius, Dinner-Diner Matching Probabilities
R. D. McKelvey and A. McLennan, The maximal number of regular totally mixed Nash equilibria, J. Economic Theory, 72:411--425, 1997.
L. I. Nicolaescu, Derangements and asymptotics of the Laplace transforms of large powers of a polynomial, New York J. Math. 10 (2004) 117-131.
S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98 (1991), 617-620.
John Riordan and N. J. A. Sloane, Correspondence, 1974
R. Vidunas, Counting derangements and Nash equilibria, arXiv preprint arXiv:1401.5400 [math.CO], 2014-2016.
Raimundas Vidunas, Counting derangements and Nash equilibria Ann. Comb. 21, No. 1, 131-152 (2017).
Wikipedia, Permutations of multisets
Index entries for sequences related to card matching

Cf. A000166, A008290, A059056-A059071, A033581, A059073.

Row n=2 of A372307.

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

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);

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 *)

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

{ 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

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

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

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).

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;

A124008 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with n-4 fixed points.

Original entry on oeis.org

9, 189, 1431, 5355, 14310, 31374, 60354, 105786, 172935, 267795, 397089, 568269

Offset: 0

Zerinvary Lajos, Nov 01 2006

nonn

			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;

A124009 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with one fixed point.

Original entry on oeis.org

0, 0, 216, 49464, 23123880, 19180338840, 25791442770240, 52614269909090064, 154809621283047068016, 631429039396055199165840, 3457808596178310768284115720, 24763433580060911383347280813320

Offset: 0

Zerinvary Lajos, Nov 01 2006

nonn

			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;

A124042 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with two fixed points.

Original entry on oeis.org

0, 9, 378, 84510, 38358540, 31234760055, 41467520432646, 83805898840005132, 244832935610272588920, 993012060508835944545045, 5413243051841698780829328690, 38622438042365626607874252846474

Offset: 0

Zerinvary Lajos, Nov 02 2006

nonn

			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;

A124043 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with three fixed points.

Original entry on oeis.org

1, 0, 435, 90944, 40563765, 32659846104, 43036380310735, 86514409614060000, 251739515511526387401, 1017865281673593548065520, 5534999211214597734889370091, 39411238922605740572075832485280

Offset: 0

Zerinvary Lajos, Nov 02 2006

nonn

			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;

A124070 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with 4 fixed points.

Original entry on oeis.org

9, 324, 69039, 30573900, 24571261710, 32346221908896, 64986793207684866, 189028409383462290696, 764111162168487304691175, 4154377697330090433618612780, 29576798800687086868033152117849

Offset: 0

Zerinvary Lajos, Nov 05 2006

nonn

			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, A124007, A124008, A124009, A124042, A124043.

cp's OEIS Frontend

A027468 9 times the triangular numbers A000217.

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

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Magma

Maple

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PARI

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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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Comments

Examples

Links

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Programs

Maple

Mathematica

Python

Formula

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

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Maple

A124008 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with n-4 fixed points.

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Maple

A124009 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with one fixed point.

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Maple

A124042 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with two fixed points.

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Maple

A124043 Number of permutations of n distinct letters (ABCD...) each of which appears thrice with three fixed points.