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 1-6 of 6 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

A177249 Number of permutations of [n] having no adjacent transpositions, that is, no cycles of the form (i, i+1).

Original entry on oeis.org

1, 1, 1, 4, 19, 99, 611, 4376, 35621, 324965, 3285269, 36462924, 440840359, 5767387591, 81184266631, 1223531387056, 19657686459529, 335404201199049, 6056933308042409, 115417137054004820, 2314399674388138811, 48717810299204919851, 1074106226256896375531
Offset: 0

Views

Author

Emeric Deutsch, May 07 2010

Keywords

Examples

			a(3)=4 because we have (1)(2)(3), (13)(2), (123), and (132).

Links

Crossrefs

Cf. A000166, A008290, A177248, A177250, A177251, A177252, A177253.

Programs

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

Formula

a(n) = A177248(n,0).
Limit_{n->oo} a(n)/n! = 1.
a(n) = Sum_{j=0..floor(n/2)} (-1)^j*(n-j)!/j!.
a(n) - n*a(n-1) = a(n-2) if n is odd.
a(n) - n*a(n-1) = a(n-2) + 2*(-1)^(n/2) if n is even.
The o.g.f. g(z) satisfies z^2*(1+z^2)*g'(z)-(1+z^2)(1-z-z^2)g(z)+1-z^2=0; g(0)=1.
The e.g.f. G(z) satisfies (1-z)G"(z)-2G'(z)-G(z)=-2cos(z); G(0)=1, G'(0)=1.
The o.g.f. is hypergeometric2F0([1,1], [], x/(1+x^2))/(1+x^2). - Mark van Hoeij, Nov 08 2011
G.f.: 1/Q(0), where Q(k)= 1 + x^2 - x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 20 2013
D-finite with recurrence a(n) = n*a(n-1) + (n-2)*a(n-3) + a(n-4). - R. J. Mathar, Jul 26 2022
G.f.: Sum_{k>=0} k! * x^k / (1+x^2)^(k+1). - Seiichi Manyama, Feb 20 2024

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

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 1-6 of 6 results.

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