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 1-10 of 10 results.

A177251 Number of permutations of [n] having no adjacent 3-cycles, i.e., no cycles of the form (i, i+1, i+2).

Original entry on oeis.org

1, 1, 2, 5, 22, 114, 697, 4923, 39612, 357899, 3588836, 39556420, 475392841, 6187284605, 86701097310, 1301467245329, 20835850494474, 354382860600678, 6381494425302865, 121290065781743383, 2426510081356069016, 50969474697328055063, 1121571023472780698152
Offset: 0

Views

Author

Emeric Deutsch, May 07 2010

Keywords

Examples

			a(4)=22 because the only permutations of {1,2,3,4} having adjacent 3-cycles are (123)(4) and (1)(234).

Links

Crossrefs

Cf. A000166, A008290, A177248, A177249, A177250, A177252, A177253.
Cf. A370324, A370569.

Programs

  • Magma
    A177251:= func< n | (&+[(-1)^j*Factorial(n-2*j)/Factorial(j): j in [0..Floor(n/3)]]) >;
[A177251(n): n in [0..30]]; // G. C. Greubel, Apr 28 2024
  • Maple
    a := proc (n) options operator, arrow: sum((-1)^j*factorial(n-2*j)/factorial(j), j = 0 .. floor((1/3)*n)) end proc: seq(a(n), n = 0 .. 22);
  • Mathematica
    a[n_] := Sum[(-1)^j*(n - 2*j)!/j!, {j, 0, n/3}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 17 2017 *)
  • SageMath
    def A177251(n): return sum((-1)^j*factorial(n-2*j)/factorial(j) for j in range(1+n//3))
[A177251(n) for n in range(31)] # G. C. Greubel, Apr 28 2024

Formula

a(n) = Sum_{j=0..floor(n/3)} (-1)^j*(n-2*j)!/j!.
a(n) = A177250(n,0).
a(n) - n*a(n-1) = 2*a(n-3) + 3*(-1)^(n/3) if 3 | n, otherwise a(n) - n*a(n-1) = 2*a(n-3).
lim_{n -> oo} a(n)/n! = 1.
The o.g.f. g(z) satisfies z^2*(1+z^3)*g'(z) - (1+z^3)(1-z-2z^3)g(z) + 1 - 2z^3 = 0; g(0)=1.
G.f.: hypergeometric2F0([1,1], [], x/(1+x^3))/(1+x^3). - Mark van Hoeij, Nov 08 2011
D-finite with recurrence a(n) = n*a(n-1) + a(n-3) + (n-3)*a(n-4) + 2*a(n-6). - R. J. Mathar, Jul 26 2022
G.f.: Sum_{k>=0} k! * x^k / (1+x^3)^(k+1). - Seiichi Manyama, Feb 20 2024

Extensions

Crossreferences corrected by Emeric Deutsch, May 09 2010

A177248 Triangle read by rows: T(n,k) is the number of permutations of [n] having k adjacent transpositions (0 <= k <= floor(n/2)). An adjacent transposition is a cycle of the form (i, i+1).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 19, 4, 1, 99, 18, 3, 611, 99, 9, 1, 4376, 612, 48, 4, 35621, 4376, 306, 16, 1, 324965, 35620, 2190, 100, 5, 3285269, 324965, 17810, 730, 25, 1, 36462924, 3285270, 162480, 5940, 180, 6, 440840359, 36462924, 1642635, 54160, 1485, 36, 1
Offset: 0

Views

Author

Emeric Deutsch, May 07 2010

Keywords

Comments

Row n contains 1 + floor(n/2) entries.
Sum of entries in row n = n! (A000142).

Examples

			T(5,2)=3 because we have (12)(34)(5), (12)(3)(45), and (1)(23)(45).
Triangle starts:
    1;
    1;
    1,  1;
    4,  2;
   19,  4,  1;
   99, 18,  3;
  611, 99,  9,  1;

Links

Crossrefs

Columns k=0..3 give A177249, A370524, A370426, A370529.
Cf. A000166, A008290, A177249, A177250, A177251, A177252, A177253.
Cf. A000142 (row sums).

Programs

  • Magma
    F:=Factorial;
A177248:= func< n,k | (&+[(-1)^j*F(n-k-j)/(F(k)*F(j)): j in [0..Floor((n-2*k)/2)]]) >;
[A177248(n,k): k in [0..Floor(n/2)], n in [0..16]]; // G. C. Greubel, Apr 28 2024
  • Maple
    T := proc (n, k) options operator, arrow: sum((-1)^(k+j)*binomial(j, k)*factorial(n-j)/factorial(j), j = 0 .. floor((1/2)*n)) end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
  • Mathematica
    T[n_, k_] := Sum[(-1)^(k + j)*Binomial[j, k]*(n - j)!/j!, {j, 0, n/2}];
Table[T[n, k], {n, 0, 12}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Nov 20 2017 *)
  • PARI
    T(n, k) = sum(j=0, n\2, (-1)^(k+j)*binomial(j,k)*(n-j)!/j!);
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ Michel Marcus, Nov 21 2017
  • SageMath
    f=factorial;
def A177248(n,k): return sum((-1)^j*f(n-k-j)/(f(k)*f(j)) for j in range(1+(n-2*k)//2))
flatten([[A177248(n,k) for k in range(1+n//2)] for n in range(17)]) # G. C. Greubel, Apr 28 2024

Formula

T(n, k) = Sum_{j=0..floor(n/2)} (-1)^(k+j)*binomial(j,k)*(n-j)!/j!.
T(n, 0) = A177249(n).
Sum_{k=0..floor(n/2)} k*T(n,k) = (n-1)! (n >= 2).
G.f. of column k: (1/k!) * Sum_{j>=k} j! * x^(j+k) / (1+x^2)^(j+1). - Seiichi Manyama, Feb 24 2024

A177250 Triangle read by rows: T(n,k) is the number of permutations of [n] having k adjacent 3-cycles (0 <= k <= floor(n/3)), i.e., having k cycles of the form (i, i+1, i+2).

Original entry on oeis.org

1, 1, 2, 5, 1, 22, 2, 114, 6, 697, 22, 1, 4923, 114, 3, 39612, 696, 12, 357899, 4923, 57, 1, 3588836, 39612, 348, 4, 39556420, 357900, 2460, 20, 475392841, 3588836, 19806, 116, 1, 6187284605, 39556420, 178950, 820, 5, 86701097310, 475392840, 1794420, 6600, 30
Offset: 0

Views

Author

Emeric Deutsch, May 07 2010

Keywords

Comments

Row n contains 1 + floor(n/3) entries.
Sum of entries in row n = n! (A000142).

Examples

			T(7,2)=3 because we have (123)(456)(7), (123)(4)(567), and (1)(234)(567).
Triangle starts:
    1;
    1;
    2;
    5,  1;
   22,  2;
  114,  6;
  697, 22,  1;

Links

Crossrefs

Columns k=0..3 give A177251, A370525, A370528, A370530.
Cf. A000142, A000166, A008290, A177248, A177249, A177252, A177253.

Programs

  • Magma
    F:=Factorial;
A177250:= func< n,k | (&+[(-1)^j*F(n-2*k-2*j)/(F(k)*F(j)): j in [0..Floor((n-3*k)/3)]]) >;
[A177250(n,k): k in [0..Floor(n/3)], n in [0..12]]; // G. C. Greubel, Apr 28 2024
  • Maple
    T := proc (n, k) options operator, arrow: sum((-1)^(k+j)*binomial(j, k)*factorial(n-2*j)/factorial(j), j = 0 .. floor((1/3)*n)) end proc: for n from 0 to 14 do seq(T(n, k), k = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form
  • Mathematica
    T[n_, k_] := Sum[(-1)^(k + j)*Binomial[j, k]*(n - 2 j)!/j!, {j, 0, n/3}];
Table[T[n, k], {n, 0, 14}, {k, 0, n/3}] // Flatten (* Jean-François Alcover, Nov 20 2017 *)
  • SageMath
    f=factorial;
def A177250(n,k): return sum((-1)^j*f(n-2*k-2*j)/(f(k)*f(j)) for j in range(1+(n-3*k)//3))
flatten([[A177250(n,k) for k in range(1+n//3)] for n in range(13)]) # G. C. Greubel, Apr 28 2024

Formula

T(n, k) = Sum_{j=0..floor(n/3)} (-1)^(k+j)*binomial(j,k)*(n-2j)!/j!.
T(n, 0) = A177251(n).
Sum_{k>=0} k*T(n,k) = (n-2)! (n>=3).
G.f. of column k: (1/k!) * Sum_{j>=k} j! * x^(j+2*k) / (1+x^3)^(j+1). - Seiichi Manyama, Feb 24 2024

Extensions

Crossreferences corrected by Emeric Deutsch, May 09 2010

A177252 Triangle read by rows: T(n,k) is the number of permutations of [n] having k adjacent 4-cycles (0 <= k <= floor(n/4)), i.e., having k cycles of the form (i, i+1, i+2, i+3).

Original entry on oeis.org

1, 1, 2, 6, 23, 1, 118, 2, 714, 6, 5016, 24, 40201, 118, 1, 362163, 714, 3, 3623772, 5016, 12, 39876540, 40200, 60, 478639079, 362163, 357, 1, 6223394516, 3623772, 2508, 4, 87138394540, 39876540, 20100, 20, 1307195547720, 478639080, 181080, 120
Offset: 0

Views

Author

Emeric Deutsch, May 07 2010

Keywords

Comments

Row n contains 1 + floor(n/4) entries.
Sum of entries in row n = n! (A000142).

Examples

			T(9,2)=3 because we have (1234)(5678)(9), (1234)(5)(6789), and (1)(2345)(6789).
Triangle starts:
     1;
     1;
     2;
     6;
    23,  1;
   118,  2;
   714,  6;
  5016, 24;

Links

Crossrefs

Columns k=0-3 give A177253, A369098, A370652, A370653.
Cf. A000166, A008290, A177248, A177249, A177250, A177251.
Cf. A000142 (row sums).

Programs

  • Magma
    A177252:= func< n,k | (&+[(-1)^j*Factorial(n-3*k-3*j)/(Factorial(k) *Factorial(j)): j in [0..Floor((n-4*k)/4)]]) >;
[A177252(n,k): k in [0..Floor(n/4)], n in [0..20]]; // G. C. Greubel, Apr 28 2024
  • Maple
    T := proc (n, k) options operator, arrow: sum((-1)^(k+j)*binomial(j, k)*factorial(n-3*j)/factorial(j), j = 0 .. floor((1/4)*n)) end proc: for n from 0 to 15 do seq(T(n, k), k = 0 .. floor((1/4)*n)) end do; % yields sequence in triangular form
  • Mathematica
    T[n_, k_]:= T[n, k]= Sum[(-1)^(k+j)*Binomial[j,k]*(n-3 j)!/j!, {j,0,n/4}];
Table[T[n, k], {n, 0, 15}, {k, 0, n/4}] // Flatten (* Jean-François Alcover, Nov 17 2017 *)
  • SageMath
    def A177252(n,k): return sum((-1)^j*factorial(n-3*k-3*j)/(factorial(k) *factorial(j)) for j in range(1+(n-4*k)//4))
flatten([[A177252(n,k) for k in range(1+n//4)] for n in range(21)]) # G. C. Greubel, Apr 28 2024

Formula

T(n,k) = Sum_{j=0..floor(n/4)} (-1)^(k+j)*binomial(j,k)*(n-3*j)!/j!.
T(n,0) = A177253(n).
Sum_{k>=0} k*T(n,k) = (n-3)! (n >= 4).
G.f. of column k: (1/k!) * Sum_{j>=k} j! * x^(j+3*k) / (1+x^4)^(j+1). - Seiichi Manyama, Feb 24 2024

A177253 Number of permutations of [n] having no adjacent 4-cycles, i.e., no cycles of the form (i, i+1, i+2, i+3).

Original entry on oeis.org

1, 1, 2, 6, 23, 118, 714, 5016, 40201, 362163, 3623772, 39876540, 478639079, 6223394516, 87138394540, 1307195547720, 20916564680761, 355600269756485, 6401066270800350, 121624180731849810, 2432546364331038479, 51084540451761077514, 1123879093137556106358
Offset: 0

Views

Author

Emeric Deutsch, May 07 2010

Keywords

Examples

			a(5)=118 because the only permutations of {1,2,3,4,5} having adjacent 4-cycles are (1234)(5) and (1)(2345).

Links

Crossrefs

Cf. A000166, A008290, A177248, A177249, A177250, A177251, A177252.

Programs

  • Magma
    [(&+[(-1)^j*Factorial(n-3*j)/Factorial(j): j in [0..Floor(n/4)]]): n in [0..30]]; // G. C. Greubel, May 11 2024
  • Maple
    a := proc (n) options operator, arrow: sum((-1)^j*factorial(n-3*j)/factorial(j), j = 0 .. floor((1/4)*n)) end proc: seq(a(n), n = 0 .. 22);
  • Mathematica
    a[n_] := Sum[(-1)^j*(n - 3*j)!/j!, {j, 0, n/4}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 20 2017 *)
  • SageMath
    [sum((-1)^j*factorial(n-3*j)/factorial(j) for j in range(1+n//4)) for n in range(31)] # G. C. Greubel, May 11 2024

Formula

a(n) = Sum_{j=0..floor(n/4)} (-1)^j*(n-3*j)!/j!.
a(n) - n*a(n-1) = 3*a(n-4) + 4*(-1)^{n/4} if 4|n otherwise a(n) - n*a(n-1) = 3*a(n-4).
a(n) = A177252(n,0).
limit_{n->oo} a(n)/n! = 1.
The o.g.f. g(z) satisfies z^2*(1+z^4)*g'(z) - (1+z^4)(1-z-3z^4)g(z) + 1 - 3z^4 = 0; g(0)=1.
D-finite with recurrence a(n) = n*a(n-1) + 2*a(n-4) + (n-4)*a(n-5) + 3*a(n-8). - R. J. Mathar, Jul 26 2022
G.f.: Sum_{k>=0} k! * x^k / (1+x^4)^(k+1). - Seiichi Manyama, Feb 20 2024

Extensions

Crossreferences corrected by Emeric Deutsch, May 09 2010

A370426 Number of permutations of [n] having exactly two adjacent 2-cycles.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 9, 48, 306, 2190, 17810, 162480, 1642635, 18231465, 220420179, 2883693792, 40592133316, 611765693532, 9828843229764, 167702100599520, 3028466654021205, 57708568527002415, 1157199837194069405, 24358905149602459920, 537053113128448187766
Offset: 0

Views

Author

Seiichi Manyama, Feb 21 2024

Keywords

Links

Crossrefs

Column k=2 of A177248.
Cf. A177249, A370524, A370529.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=2, N, k!*x^(k+2)/(1+x^2)^(k+1))/2))
  • PARI
    a(n, k=2, q=2) = sum(j=0, n\q-k, (-1)^j*(n-(q-1)*(j+k))!/j!)/k!;

Formula

G.f.: (1/2) * Sum_{k>=2} k! * x^(k+2) / (1+x^2)^(k+1).
a(n) = (1/2) * Sum_{k=0..floor(n/2)-2} (-1)^k * (n-k-2)! / k!.
a(n) ~ n! / (2*n^2). - Vaclav Kotesovec, May 23 2025

A370524 Number of permutations of [n] having exactly one adjacent 2-cycle.

Original entry on oeis.org

0, 0, 1, 2, 4, 18, 99, 612, 4376, 35620, 324965, 3285270, 36462924, 440840358, 5767387591, 81184266632, 1223531387056, 19657686459528, 335404201199049, 6056933308042410, 115417137054004820, 2314399674388138810, 48717810299204919851, 1074106226256896375532
Offset: 0

Views

Author

Seiichi Manyama, Feb 21 2024

Keywords

Examples

			The permutations of {1,2,3} having exactly one adjacent 2-cycle are (12)(3) and (1)(23). So a(3) = 2.

Links

Crossrefs

Column k=2 of A370527.
Column k=1 of A177248
Cf. A177249, A370426, A370529.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, k!*x^(k+1)/(1+x^2)^(k+1))))
  • PARI
    a(n, k=1, q=2) = sum(j=0, n\q-k, (-1)^j*(n-(q-1)*(j+k))!/j!)/k!;

Formula

G.f.: Sum_{k>=1} k! * x^(k+1) / (1+x^2)^(k+1).
a(n) = Sum_{k=0..floor(n/2)-1} (-1)^k * (n-k-1)! / k!.

A370529 Number of permutations of [n] having exactly three adjacent 2-cycles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 4, 16, 100, 730, 5940, 54160, 547540, 6077155, 73473400, 961231264, 13530711096, 203921897844, 3276281076600, 55900700199840, 1009488884673720, 19236189509000805, 385733279064689820, 8119635049867486640, 179017704376149395900
Offset: 0

Views

Author

Seiichi Manyama, Feb 21 2024

Keywords

Links

Crossrefs

Column k=3 of A177248.
Cf. A177249, A370426, A370524.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<7, [0$6, 1][n+1], ((n-5)*(n-6)*(n-3)*a(n-1)
       -6*(n-4)*a(n-2)+(n-2)*(n-3)*((n-5)*a(n-3)+a(n-4)))/((n-5)*(n-6)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 21 2024
  • PARI
    my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0], Vec(sum(k=3, N, k!*x^(k+3)/(1+x^2)^(k+1))/6))
  • PARI
    a(n, k=3, q=2) = sum(j=0, n\q-k, (-1)^j*(n-(q-1)*(j+k))!/j!)/k!;

Formula

G.f.: (1/6) * Sum_{k>=3} k! * x^(k+3) / (1+x^2)^(k+1).
a(n) = (1/6) * Sum_{k=0..floor(n/2)-3} (-1)^k * (n-k-3)! / k!.
a(n) ~ n! / (6*n^3). - Vaclav Kotesovec, Feb 21 2024

A370669 Expansion of Sum_{k>=0} k! * ( x/(1+x^2) )^k.

Original entry on oeis.org

1, 1, 2, 5, 20, 103, 630, 4475, 36232, 329341, 3320890, 36787889, 444125628, 5803850515, 81625106990, 1229298774647, 19738870726160, 336627732586105, 6076590994501938, 115752541255203869, 2320456607696181220, 48833227436258924671, 1076420625931284514342
Offset: 0

Views

Author

Seiichi Manyama, Feb 25 2024

Keywords

Crossrefs

Cf. A000255, A370670.
Cf. A177249, A212580, A276080.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x/(1+x^2))^k))
  • PARI
    a(n) = sum(k=0, n\2, (-1)^k*(n-2*k)!*binomial(n-k-1, k));

Formula

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * (n-2*k)! * binomial(n-k-1,k).
a(n) = n*a(n-1) + (n-4)*a(n-3) + a(n-4) for n > 4.

A370674 Expansion of Sum_{k>0} k! * x^k/(1+x^k)^(k+1).

Original entry on oeis.org

1, 0, 9, 14, 125, 702, 5047, 40172, 362949, 3628100, 39916811, 478996746, 6227020813, 87178250922, 1307674370745, 20922789524192, 355687428096017, 6402373702119096, 121645100408832019, 2432902008136718030, 51090942171709621965, 1124000727777128678510
Offset: 1

Views

Author

Seiichi Manyama, Feb 25 2024

Keywords

Crossrefs

Cf. A177249, A177251, A177253.
Cf. A344777, A358411.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, k!*x^k/(1+x^k)^(k+1)))
  • PARI
    a(n) = sumdiv(n, d, (-1)^(d-1)*(d+n/d-1)!/(d-1)!);

Formula

a(n) = Sum_{d|n} (-1)^(d-1) * (d+n/d-1)!/(d-1)!.
If p is an odd prime, a(p) = p + p!.
Showing 1-10 of 10 results.

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