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.

Previous Showing 21-27 of 27 results.

A336174 Number of non-symmetric binary n X n matrices M over the reals such that M^2 is the transpose of M.

Original entry on oeis.org

0, 0, 0, 2, 16, 80, 360, 1680, 8064, 39872, 209920, 1168640, 6779520, 41403648, 265434624, 1765487360, 12227461120, 88163164160, 656547803136, 5054718763008, 40261284495360, 330010833797120, 2783003768258560, 24166721457815552, 215318925878132736, 1966855934150246400
Offset: 0

Views

Author

Torlach Rush, Jul 10 2020

Keywords

Comments

We classify the (0,1) n X n matrices M_n by k, the number of 1's.
Let [T(n,k), n >= 0, k=0..n], be the lower triangular matrix where T(n,k) is the number of M^2 matrices equal to the transpose of M for n and k. Then:
T(n,n) = A001471(n).
Column sequences k=3..7 (without leading 0's) are:
T(n,3) = A001471(3) * A000292(n+1).
T(n,4) = A001471(4) * A000332(n+4).
T(n,5) = A001471(5) * A000389(n+5).
T(n,6) = A001471(6) * A000579(n+6).
T(n,7) = A001471(7) * A000580(n+7).
Row sums of T(n,k) generate known terms of this sequence and the next term a(10) evaluates to 209920 (see conjectured formula below).

Examples

			a(3) = 2 because [0,1,0]    [0,1,0]    [0,0,1]
                 [0,0,1]  * [0,0,1]  = [1,0,0]
                 [1,0,0]    [1,0,0]    [0,1,0],
             and [0,0,1]    [0,0,1]    [0,1,0]
                 [1,0,0]  * [1,0,0]  = [0,0,1]
                 [0,1,0]    [0,1,0]    [1,0,0].

Crossrefs

Cf. A000292, A000332, A000389, A000579, A000580, A001471, A344912.

Programs

  • Maple
    a := n -> 2^n*(add(n!/(24^k*k!*(n-3*k)!), k=0..n/3) - 1): seq(a(n), n=0..25);
# Alternative:
gf := exp(x*(x^2+6)/3) - exp(2*x): ser := series(gf,x,32):
seq(n!*coeff(ser,x,n), n = 0..25); # Peter Luschny, Jun 05 2021
  • PARI
    m(n, t) = matrix(n, n, i, j, (t>>(i*n+j-n-1))%2)
a(n) = sum(t = 0, 2^n^2-1, m(n, t)^2 == m(n, t)~) - 2^n
for(n = 0, 9, print1(a(n), ", "))

Formula

a(n) = Sum_{k=0..n} A001471(k) * binomial(n, k). [Previously conjectured, for a proof see the link in A344912.]
From Peter Luschny, Jun 05 2021: (Start)
a(n) = 2^n*(add(n!/(24^k * k! * (n - 3*k)!), k=0..n/3) - 1).
a(n) = 2^n*(hypergeom([-n/3, (1 - n)/3, (2 - n)/3], [], -9/8) - 1).
a(n) = [x^n] exp(x*(x^2 + 6)/3) - exp(2*x). (End)
D-finite with recurrence (-n+3)*a(n) +4*(n-2)*a(n-1) +4*(-n+1)*a(n-2) +(n-1)*(n-2)*(n-3)*a(n-3) -2*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Jul 27 2022

Extensions

More terms from Peter Luschny, Jun 05 2021

A061122 Number of degree-n permutations of order exactly 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 5040, 45360, 453600, 3326400, 39916800, 363242880, 3874590720, 34767532800, 567177811200, 6897521030400, 98241008785920, 1138935652807680, 18952720774041600, 258251731634534400
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.

Formula

E.g.f.: -exp(x+1/2*x^2+1/4*x^4)+exp(x+1/2*x^2+1/4*x^4+1/8*x^8).

A061123 Number of degree-n permutations of order exactly 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 40320, 403200, 2217600, 26611200, 259459200, 1695133440, 16345929600, 161902540800, 1208560953600, 50830132953600, 866513503215360, 8470676211379200, 166891791625977600, 2699606616475507200
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.

Formula

E.g.f.: -exp(x+1/3*x^3)+exp(x+1/3*x^3+1/9*x^9).

A061124 Number of degree-n permutations of order exactly 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 504, 4032, 27216, 514080, 4823280, 57081024, 500972472, 4412103696, 60619398840, 686638592640, 9335025764064, 104304736815552, 1180585704051936, 29016515871665280, 478096386437121480
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.

Formula

E.g.f.: exp(x) - exp(x+1/2*x^2) - exp(x+1/5*x^5) + exp(x+1/2*x^2+1/5*x^5+1/10*x^10).
From Benedict W. J. Irwin, May 27 2016: (Start)
Let y1(0)=1, y1(1)=1,
Let -y1(n)-y1(n+1)+(n+2)*y1(n+2)=0,
Let y2(0)=1, y2(1)=1, y2(2)=1/2, y2(3)=1/6, y2(4)=1/24,
Let -y2(n)-y2(n+4)+(n+5)*y2(n+5)=0,
Let y3(0)=1, y3(1)=1, y3(2)=1, y3(3)=2/3, y3(4)=5/12, y3(5)=5/12, y3(6)=11/36, y3(7)=31/126, y3(8)=307/2016, y3(9)=1643/18144,
Let -y3(n)-y3(n+5)-y3(n+8)-y3(n+9)+(n+10)*y3(n+10)=0,
a(n) = 1+n!*(y3(n)-y2(n)-y1(n)).
(End)

A061125 Number of degree-n permutations of order exactly 12.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 420, 3360, 30240, 403200, 4019400, 80166240, 965284320, 12173441280, 162850287600, 2428557331200, 32123543612160, 534678700308480, 8126981741380320, 128338880777251200, 2080312367956502400, 36351373041072122880, 606331931399062693440
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128.

Programs

  • Mathematica
    nn=21;Range[0,nn]!CoefficientList[Series[(Exp[x^12/12]-1)Exp[x+x^2/2+x^3/3+x^4/4+x^6/6]+(Exp[x^6/6]-1)(Exp[x^4/4]-1)Exp[x+x^2/2+x^3/3]+(Exp[x^4/4]-1)(Exp[x^3/3]-1)Exp[x^2/2+x],{x,0,nn}],x]//Rest  (* Geoffrey Critzer, Feb 04 2013 *)

Formula

E.g.f.: exp(x + 1/2*x^2) - exp(x + 1/2*x^2 + 1/4*x^4) - exp(x + 1/2*x^2 + 1/3*x^3 + 1/6*x^6) + exp(x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 + 1/6*x^6 + 1/12*x^12).

A061126 Number of degree-n permutations of order exactly 16.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1307674368000, 22230464256000, 400148356608000, 5068545850368000, 101370917007360000, 1490152480008192000, 24977793950613504000, 343667682838351872000
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128.

Formula

E.g.f.: - exp(x + 1/2*x^2 + 1/4*x^4 + 1/8*x^8) + exp(x + 1/2*x^2 + 1/4*x^4 + 1/8*x^8 + 1/16*x^16).

A061127 Number of degree-n permutations of order exactly 24.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1663200, 19958400, 259459200, 4843238400, 72648576000, 988020633600, 14600749363200, 224704121241600, 3614691131251200, 84808750650624000, 1509309706083379200, 29359195162807910400
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128.

Programs

  • Mathematica
    nn=22;Range[0,nn]!CoefficientList[Series[(Exp[x^24/24]-1)Exp[x+x^2/2+x^3/3+x^4/4+x^6/6+x^8/8+x^12/12]+(Exp[x^12/12]-1)(Exp[x^8/8]-1)Exp[x+x^2/2+x^3/3+x^4/4+x^6/6]+(Exp[x^8/8]-1)(Exp[x^6/6]-1)Exp[x+x^2/2+x^3/3+x^4/4]+(Exp[x^8/8]-1)(Exp[x^3/3]-1)Exp[x+x^2/2+x^4/4],{x,0,nn}],x]//Rest (* Geoffrey Critzer, Feb 04 2013 *)

Formula

E.g.f.: exp(x + 1/2*x^2 + 1/4*x^4) - exp(x + 1/2*x^2 + 1/4*x^4 + 1/8*x^8) - exp(x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 + 1/6*x^6 + 1/12*x^12) + exp(x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 + 1/6*x^6 + 1/8*x^8 + 1/12*x^12 + 1/24*x^24).
Previous Showing 21-27 of 27 results.

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