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 1-5 of 5 results.

A000172 The Franel number a(n) = Sum_{k = 0..n} binomial(n,k)^3.

Original entry on oeis.org

1, 2, 10, 56, 346, 2252, 15184, 104960, 739162, 5280932, 38165260, 278415920, 2046924400, 15148345760, 112738423360, 843126957056, 6332299624282, 47737325577620, 361077477684436, 2739270870994736, 20836827035351596, 158883473753259752, 1214171997616258240
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Cusick gives a general method of deriving recurrences for the r-th order Franel numbers (this is the sequence of third-order Franel numbers), with floor((r+3)/2) terms.
This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004
An identity of V. Strehl states that a(n) = Sum_{k = 0..n} C(n,k)^2 * binomial(2*k,n). Zhi-Wei Sun conjectured that for every n = 2,3,... the polynomial f_n(x) = Sum_{k = 0..n} binomial(n,k)^2 * binomial(2*k,n) * x^(n-k) is irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 21 2013
Conjecture: a(n) == 2 (mod n^3) iff n is prime. - Gary Detlefs, Mar 22 2013
a(p) == 2 (mod p^3) for any prime p since p | C(p,k) for all k = 1,...,p-1. - Zhi-Wei Sun, Aug 14 2013
a(n) is the maximal number of totally mixed Nash equilibria in games of 3 players, each with n+1 pure options. - Raimundas Vidunas, Jan 22 2014
This is one of the Apéry-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
Diagonal of rational functions 1/(1 - x*y - y*z - x*z - 2*x*y*z), 1/(1 - x - y - z + 4*x*y*z), 1/(1 + y + z + x*y + y*z + x*z + 2*x*y*z), 1/(1 + x + y + z + 2*(x*y + y*z + x*z) + 4*x*y*z). - Gheorghe Coserea, Jul 04 2018
a(n) is the constant term in the expansion of ((1 + x) * (1 + y) + (1 + 1/x) * (1 + 1/y))^n. - Seiichi Manyama, Oct 27 2019
Diagonal of rational function 1 / ((1-x)*(1-y)*(1-z) - x*y*z). - Seiichi Manyama, Jul 11 2020
Named after the Swiss mathematician Jérôme Franel (1859-1939). - Amiram Eldar, Jun 15 2021
It appears that a(n) is equal to the coefficient of (x*y*z)^n in the expansion of (1 + x + y - z)^n * (1 + x - y + z)^n * (1 - x + y + z)^n. Cf. A036917. - Peter Bala, Sep 20 2021

Examples

			O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 56*x^3 + 346*x^4 + 2252*x^5 + ...
O.g.f.: A(x) = 1/(1-2*x) + 3!*x^2/(1-2*x)^4 + (6!/2!^3)*x^4/(1-2*x)^7 + (9!/3!^3)*x^6/(1-2*x)^10 + (12!/4!^3)*x^8/(1-2*x)^13 + ... - _Paul D. Hanna_, Oct 30 2010
Let g.f. A(x) = Sum_{n >= 0} a(n)*x^n/n!^3, then
A(x) = 1 + 2*x + 10*x^2/2!^3 + 56*x^3/3!^3 + 346*x^4/4!^3 + ... where
A(x) = [1 + x + x^2/2!^3 + x^3/3!^3 + x^4/4!^3 + ...]^2. - _Paul D. Hanna_

References

  • Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
  • Jérôme Franel, On a question of Laisant, Intermédiaire des Mathématiciens, vol 1 1894 pp 45-47
  • H. W. Gould, Combinatorial Identities, Morgantown, 1972, (X.14), p. 56.
  • Murray Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 148-149.
  • John 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. A002893, A052144, A005260, A096191, A033581, A189791. Second row of array A094424.
Cf. A181543, A006480, A141057, A000578, A007318.
The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.
Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
Column k=3 of A372307.

Programs

  • Haskell
    a000172 = sum . map a000578 . a007318_row
-- Reinhard Zumkeller, Jan 06 2013
  • Maple
    A000172 := proc(n)
    add(binomial(n,k)^3,k=0..n) ;
end proc:
seq(A000172(n),n=0..10) ; # R. J. Mathar, Jul 26 2014
A000172_list := proc(len) series(hypergeom([], [1, 1], x)^2, x, len);
seq((n!)^3*coeff(%, x, n), n=0..len-1) end:
A000172_list(21); # Peter Luschny, May 31 2017
  • Mathematica
    Table[Sum[Binomial[n,k]^3,{k,0,n}],{n,0,30}] (* Harvey P. Dale, Aug 24 2011 *)
Table[ HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -1], {n, 0, 20}]  (* Jean-François Alcover, Jul 16 2012, after symbolic sum *)
a[n_] := Sum[ Binomial[2k, n]*Binomial[2k, k]*Binomial[2(n-k), n-k], {k, 0, n}]/2^n; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 20 2013, after Zhi-Wei Sun *)
a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/3, 2/3, 1, 27 x^2 / (1 - 2 x)^3] / (1 - 2 x), {x, 0, n}]; (* Michael Somos, Jul 16 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0,n,(3*m)!/m!^3*x^(2*m)/(1-2*x+x*O(x^n))^(3*m+1)),n)} \\ Paul D. Hanna, Oct 30 2010
  • PARI
    {a(n)=n!^3*polcoeff(sum(m=0,n,x^m/m!^3+x*O(x^n))^2,n)} \\ Paul D. Hanna, Jan 19 2011
  • PARI
    A000172(n)={sum(k=0,(n-1)\2,binomial(n,k)^3)*2+if(!bittest(n,0),binomial(n,n\2)^3)} \\ M. F. Hasler, Sep 21 2015
  • Sage
    def A000172():
    x, y, n = 1, 2, 1
    while True:
        yield x
        n += 1
        x, y = y, (8*(n-1)^2*x + (7*n^2-7*n + 2)*y) // n^2
a = A000172()
[next(a) for i in range(21)]   # Peter Luschny, Oct 12 2013

Formula

A002893(n) = Sum_{m = 0..n} binomial(n, m)*a(m) [Barrucand].
Sum_{k = 0..n} C(n, k)^3 = (-1)^n*Integral_{x = 0..infinity} L_k(x)^3 exp(-x) dx. - from Askey's book, p. 43.
D-finite with recurrence (n + 1)^2*a(n+1) = (7*n^2 + 7*n + 2)*a(n) + 8*n^2*a(n-1) [Franel]. - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jan 31 2001
a(n) ~ 2*3^(-1/2)*Pi^-1*n^-1*2^(3*n). - Joe Keane (jgk(AT)jgk.org), Jun 21 2002
O.g.f.: A(x) = Sum_{n >= 0} (3*n)!/n!^3 * x^(2*n)/(1 - 2*x)^(3*n+1). - Paul D. Hanna, Oct 30 2010
G.f.: hypergeom([1/3, 2/3], [1], 27 x^2 / (1 - 2x)^3) / (1 - 2x). - Michael Somos, Dec 17 2010
G.f.: Sum_{n >= 0} a(n)*x^n/n!^3 = [ Sum_{n >= 0} x^n/n!^3 ]^2. - Paul D. Hanna, Jan 19 2011
G.f.: A(x) = 1/(1-2*x)*(1+6*(x^2)/(G(0)-6*x^2)),
with G(k) = 3*(x^2)*(3*k+1)*(3*k+2) + ((1-2*x)^3)*((k+1)^2) - 3*(x^2)*((1-2*x)^3)*((k+1)^2)*(3*k+4)*(3*k+5)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 03 2011
In 2011 Zhi-Wei Sun found the formula Sum_{k = 0..n} C(2*k,n)*C(2*k,k)*C(2*(n-k),n-k) = (2^n)*a(n) and proved it via the Zeilberger algorithm. - Zhi-Wei Sun, Mar 20 2013
0 = a(n)*(a(n+1)*(-2048*a(n+2) - 3392*a(n+3) + 768*a(n+4)) + a(n+2)*(-1280*a(n+2) - 2912*a(n+3) + 744*a(n+4)) + a(n+3)*(+288*a(n+3) - 96*a(n+4))) + a(n+1)*(a(n+1)*(-704*a(n+2) - 1232*a(n+3) + 288*a(n+4)) + a(n+2)*(-560*a(n+2) - 1372*a(n+3) + 364*a(n+4)) + a(n+3)*(+154*a(n+3) - 53*a(n+4))) + a(n+2)*(a(n+2)*(+24*a(n+2) + 70*a(n+3) - 20*a(n+4)) + a(n+3)*(-11*a(n+3) + 4*a(n+4))) for all n in Z. - Michael Somos, Jul 16 2014
For r a nonnegative integer, Sum_{k = r..n} C(k,r)^3*C(n,k)^3 = C(n,r)^3*a(n-r), where we take a(n) = 0 for n < 0. - Peter Bala, Jul 27 2016
a(n) = (n!)^3 * [x^n] hypergeom([], [1, 1], x)^2. - Peter Luschny, May 31 2017
From Gheorghe Coserea, Jul 04 2018: (Start)
a(n) = Sum_{k=0..floor(n/2)} (n+k)!/(k!^3*(n-2*k)!) * 2^(n-2*k).
G.f. y=A(x) satisfies: 0 = x*(x + 1)*(8*x - 1)*y'' + (24*x^2 + 14*x - 1)*y' + 2*(4*x + 1)*y. (End)
a(n) = [x^n] (1 - x^2)^n*P(n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. See Gould, p. 56. - Peter Bala, Mar 24 2022
a(n) = (2^n/(4*Pi^2)) * Integral_{x,y=0..2*Pi} (1+cos(x)+cos(y)+cos(x+y))^n dx dy = (8^n/(Pi^2)) * Integral_{x,y=0..Pi} (cos(x)*cos(y)*cos(x+y))^n dx dy (Pla, 1995). - Amiram Eldar, Jul 16 2022
a(n) = Sum_{k = 0..n} m^(n-k)*binomial(n,k)*binomial(n+2*k,n)*binomial(2*k,k) at m = -4. Cf. A081798 (m = 1), A006480 (m = 0), A124435 (m = -1), A318109 (m = -2) and A318108 (m = -3). - Peter Bala, Mar 16 2023
From Bradley Klee, Jun 05 2023: (Start)
The g.f. T(x) obeys a period-annihilating ODE:
0=2*(1 + 4*x)*T(x) + (-1 + 14*x + 24*x^2)*T'(x) + x*(1 + x)*(-1 + 8*x)*T''(x).
The periods ODE can be derived from the following Weierstrass data:
g2 = (4/243)*(1 - 8*x + 240*x^2 - 464*x^3 + 16*x^4);
g3 = -(8/19683)*(1 - 12*x - 480*x^2 + 3080*x^3 - 12072*x^4 + 4128*x^5 +
64*x^6);
which determine an elliptic surface with four singular fibers. (End)
From Peter Bala, Oct 31 2024: (Start)
For n >= 1, a(n) = 2 * Sum_{k = 0..n-1} binomial(n, k)^2 * binomial(n-1, k). Cf. A361716.
For n >= 1, a(n) = 2 * hypergeom([-n, -n, -n + 1], [1, 1], -1). (End)

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

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

A375694 Number A(n,k) of multiset permutations of {{1}^k, {2}^k, ..., {n}^k} with no fixed k-tuple {j}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 5, 2, 0, 1, 0, 19, 74, 9, 0, 1, 0, 69, 1622, 2193, 44, 0, 1, 0, 251, 34442, 362997, 101644, 265, 0, 1, 0, 923, 756002, 62924817, 166336604, 6840085, 1854, 0, 1, 0, 3431, 17150366, 11729719509, 305225265804, 136221590695, 630985830, 14833, 0
Offset: 0

Views

Author

Alois P. Heinz, Aug 24 2024

Keywords

Examples

			A(2,2) = 5: 1212, 1221, 2112, 2121, 2211.
A(2,3) = 19: 112122, 112212, 112221, 121122, 121212, 121221, 122112, 122121, 122211, 211122, 211212, 211221, 212112, 212121, 212211, 221112, 221121, 221211, 222111.
A(3,2) = 74: 121323, 121332, 122313, 122331, 123123, 123132, 123213, 123231, 123312, 123321, 131223, 131232, 131322, 132123, 132132, 132312, 132321, 133122, 133212, 133221, 211323, 211332, 212313, 212331, 213123, 213132, 213213, 213231, 213312, 213321, 221313, 221331, 223113, 223131, 223311, 231123, 231132, 231213, 231231, 231312, 231321, 232113, 232131, 232311, 233112, 233121, 233211, 311223, 311232, 311322, 312123, 312132, 312312, 312321, 313122, 313212, 313221, 321123, 321132, 321213, 321231, 321312, 321321, 322113, 322131, 322311, 323112, 323121, 323211, 331122, 331212, 331221, 332112, 332121.
A(4,1) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
Square array A(n,k) begins:
  1,  1,      1,         1,            1,               1, ...
  0,  0,      0,         0,            0,               0, ...
  0,  1,      5,        19,           69,             251, ...
  0,  2,     74,      1622,        34442,          756002, ...
  0,  9,   2193,    362997,     62924817,     11729719509, ...
  0, 44, 101644, 166336604, 305225265804, 623302086965044, ...

Links

Crossrefs

Columns k=0-2 give: A000007, A000166, A374980.
Rows n=0-2 give: A000012, A000004, A030662.
Main diagonal gives A375693.
Cf. A060538, A089759, A187783, A372307.

Programs

  • Maple
    A:= (n, k)-> add((-1)^(n-j)*binomial(n, j)*(k*j)!/k!^j, j=0..n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Formula

A(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(k*j)!/k!^j.

A375778 Number of multiset permutations of {{1}^n, {2}^n, ..., {n}^n} with no fixed points.

Original entry on oeis.org

1, 0, 1, 56, 748521, 2671644472544, 4165949769769961828425, 4198063809561374304863859278045664, 3792196761630663647644729491214828673313675120817, 4066336002770373278819438293126282402268149361469069425278228336256
Offset: 0

Views

Author

Alois P. Heinz, Aug 27 2024

Keywords

Examples

			a(0) = 1: the empty permutation.
a(2) = 1: 2211.
a(3) = 56: 222333111, 223133112, 223133121, 223133211, 223313112, 223313121, 223313211, 223331112, 223331121, 223331211, 232133112, 232133121, 232133211, 232313112, 232313121, 232313211, 232331112, 232331121, 232331211, 233113122, 233113212, 233113221, 233131122, 233131212, 233131221, 233311122, 233311212, 233311221, 322133112, 322133121, 322133211, 322313112, 322313121, 322313211, 322331112, 322331121, 322331211, 323113122, 323113212, 323113221, 323131122, 323131212, 323131221, 323311122, 323311212, 323311221, 332113122, 332113212, 332113221, 332131122, 332131212, 332131221, 332311122, 332311212, 332311221, 333111222.

Links

Crossrefs

Main diagonal of A372307.
Cf. A059841, A375693.

Programs

  • Maple
    a:= n-> (-1)^n*int(exp(-x)*orthopoly[L](n, x)^n, x=0..infinity):
seq(a(n), n=0..10);
  • Mathematica
    a[n_] := (-1)^n*Integrate[Exp[-x]*LaguerreL[n, x]^n, {x, 0, Infinity}];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Sep 02 2024, after Alois P. Heinz *)

Formula

a(n) mod 2 = 1 - (n mod 2) = A059841(n).
Showing 1-5 of 5 results.

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

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