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 1-4 of 4 results.

A161119 Triangle read by rows: T(n,k) is the number of fixed-point-free involutions of {1,2,...,2n} having k cycles with entries of opposite parities (0 <= k <= n).

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 9, 0, 6, 9, 0, 72, 0, 24, 0, 225, 0, 600, 0, 120, 225, 0, 4050, 0, 5400, 0, 720, 0, 11025, 0, 66150, 0, 52920, 0, 5040, 11025, 0, 352800, 0, 1058400, 0, 564480, 0, 40320, 0, 893025, 0, 9525600, 0, 17146080, 0, 6531840, 0, 362880, 893025, 0, 44651250, 0, 238140000, 0, 285768000, 0, 81648000, 0, 3628800
Offset: 0

Views

Author

Emeric Deutsch, Jun 02 2009

Keywords

Comments

T(n,k) is the number of basis elements in the order-n Brauer algebra that have propagation number k. - John M. Campbell, Dec 08 2021

Examples

			T(3,1)=9 because we have (12)(35)(46), (14)(26)(35), (16)(24)(35), (23)(15)(46), (25)(13)(46), (34)(15)(26), (36)(15)(24), (45)(13)(26), (56)(13)(24).
Triangle starts:
  1;
  0,  1;
  1,  0,  2;
  0,  9,  0,  6;
  9,  0, 72,  0, 24;

Links

Crossrefs

Cf. A000142, A001147, A001818, A161120, A161121, A161122.

Programs

  • Maple
    T := proc (n, k) if `mod`(n-k, 2) = 1 then 0 else binomial(n, k)^2*factorial(k)*(product(2*j-1, j = 1 .. (1/2)*n-(1/2)*k))^2 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form
  • PARI
    dfo(n) = if (n<0, (-1)^n/dfo(-n), (2*n)! / n! / 2^n); \\ A001147
T(n,k) = if ((n-k)%2, 0, k!*binomial(n,k)^2*dfo((n-k)/2)^2);
row(n) = vector(n+1, k, T(n,k-1)) \\ Michel Marcus, Dec 09 2021

Formula

T(n,k) = k!*binomial(n,k)^2*(n-k-1)!!^2 if n-k is even; T(n,k) = 0 if n-k is odd.
Sum of row n = (2n-1)!! = A001147(n).
T(n,n) = n! = A000142(n).
T(2n,0) = A001818(n).
Sum_{k>=0} k*T(n,k) = n^2*(2n-3)!! = A161120(n).

A161121 Triangle read by rows: T(n,k) is the number of fixed-point-free involutions of {1,2,...,2n} having k cycles with entries of the same parity (0 <= k <= 2*floor(n/2)).

Original entry on oeis.org

1, 1, 2, 0, 1, 6, 0, 9, 24, 0, 72, 0, 9, 120, 0, 600, 0, 225, 720, 0, 5400, 0, 4050, 0, 225, 5040, 0, 52920, 0, 66150, 0, 11025, 40320, 0, 564480, 0, 1058400, 0, 352800, 0, 11025, 362880, 0, 6531840, 0, 17146080, 0, 9525600, 0, 893025, 3628800, 0, 81648000, 0
Offset: 0

Views

Author

Emeric Deutsch, Jun 02 2009

Keywords

Comments

Row n contains 1 + 2*floor(n/2) terms.
Sum of row n = (2n-1)!! (A001147).
a(n,0) = n! (A000142).
a(2n,2n) = A001818(n).
Sum_{k>=0} k*T(n,k) = n*(n-1)*(2n-3)!! = A161122(n).

Examples

			T(3,2)=9 because we have (12)(35)(46), (14)(26)(35), (16)(24)(35), (23)(15)(46), (25)(13)(46), (34)(15)(26), (36)(15)(24), (45)(13)(26), (56)(13)(24).
Triangle starts:
    1;
    1;
    2,   0,   1;
    6,   0,   9;
   24,   0,  72,   0,   9;
  120,   0, 600,   0, 225;

Links

Crossrefs

Cf. A000142, A001147, A001818, A161119, A161120, A161122.

Programs

  • Maple
    T := proc (n, k) if n < k then 0 elif `mod`(k, 2) = 0 then binomial(n, k)^2*factorial(n-k)*(product(2*j-1, j = 1 .. (1/2)*k))^2 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. 2*floor((1/2)*n)) end do; # yields sequence in triangular form
  • Mathematica
    T[n_, k_] := If[EvenQ[k], (n-k)! Binomial[n, k]^2 ((k-1)!!)^2, 0];
Table[T[n, k], {n, 0, 10}, {k, 0, 2 Quotient[n, 2]}] // Flatten (* Jean-François Alcover, Feb 01 2023 *)

Formula

T(n,k) = (n-k)!*binomial(n,k)^2*((k-1)!!)^2 if k is even; T(n,k) = 0 if k is odd.

A161122 Number of cycles with entries of the same parity in all fixed-point-free involutions of {1,2,...,2n}.

Original entry on oeis.org

0, 0, 2, 18, 180, 2100, 28350, 436590, 7567560, 145945800, 3101348250, 72020198250, 1814908995900, 49332526343100, 1438865351673750, 44826189802143750, 1485668004871050000, 52196469237802890000, 1937793920453432291250, 75801938653031321981250, 3116301922402398792562500
Offset: 0

Views

Author

Emeric Deutsch, Jun 02 2009

Keywords

Examples

			a(2)=2 because in the 3 permutations (12)(34), (13)(24), (14)(23) we have a total of 2 cycles with entries of the same parity.

Links

Crossrefs

Cf. A161119, A161120, A161121.

Programs

  • Magma
    DoubleFactorial:=func< n | &*[n..2 by -2] >; [ n*(n-1)*DoubleFactorial(2*n-3): n in [0..22]]; // Vincenzo Librandi, Jul 21 2017
  • Maple
    seq(n*(n-1)*(product(2*j-1, j = 1 .. n-1)), n = 0 .. 18);
  • Mathematica
    Table[n (n - 1) (2 n -3)!!, {n, 0, 20}] (* Vincenzo Librandi, Jul 21 2017 *)

Formula

a(n) = n(n-1)(2n-3)!!.
a(n) = Sum_{k>=0} k*A161121(n,k).
D-finite with recurrence (-n+2)*a(n) +n*(2*n-3)*a(n-1)=0. - R. J. Mathar, Jul 26 2022

A368235 Triangle read by rows: n-th row polynomial equals the numerator of the rational function (-1)^n*f(x) * (d/dx)^n (1/f(x)), where f(x) = sqrt(x + x^2).

Original entry on oeis.org

1, 1, 2, 3, 8, 8, 15, 54, 72, 48, 105, 480, 864, 768, 384, 945, 5250, 12000, 14400, 9600, 3840, 10395, 68040, 189000, 288000, 259200, 138240, 46080, 135135, 1018710, 3333960, 6174000, 7056000, 5080320, 2257920, 645120, 2027025, 17297280, 65197440, 142248960, 197568000, 180633600, 108380160, 41287680, 10321920
Offset: 0

Views

Author

Peter Bala, Dec 18 2023

Keywords

Comments

Unsigned row reverse of A123516.
The row polynomials also occur on repeated integration of 1/sqrt(x + x^2). See the example section.

Examples

			Triangle begins
 n\k |     0       1        2        3        4        5       6
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  0  |     1
  1  |     1       2
  2  |     3       8        8
  3  |    15      54       72       48
  4  |   105     480      864      768      384
  5  |   945    5250    12000    14400     9600     3840
  6  | 10395   68040   189000   288000   259200   138240   46080
 ...
Repeated integration of 1/f(x), where f(x) = sqrt(x + x^2):
Let I denote the integral operator h(x) -> Integral_{t = 0..x} h(t) dt.
Let g(x) = I(1/f(x)) = log(2*x + 1 + 2*f(x)). Then
(2^1 * 1!^2) * I^(2)(1/f(x)) = (2*x + 1)*g(x) - 2*f(x).
(2^2 * 2!^2) * I^(3)(1/f(x)) = (8*x^2 + 8*x + 3)*g(x) - 6*(2*x + 1)*f(x).
(2^3 * 3!^2) * I^(4)(1/f(x)) = (48*x^3 + 72*x^2 + 54*x + 15)*g(x) - 2*(44*x^2 + 44*x + 15)*f(x).
(2^4 * 4!^2) * I^(5)(1/f(x)) = (384*x^4 + 768*x^3 + 864*x^2 + 480*x + 105)*g(x) - 10*(2*x + 1)*(40*x^2 + 40*x + 21)*f(x).

Crossrefs

Cf. A001147 (column 1), 2*A161120 (column 2), A000165 (main diagonal) A014479 (first subdiagonal), 3*A286725 (second subdiagonal).
Cf. A059366, A098461, A123516, A156919, A177145, A286724, A331817.

Programs

  • Maple
    # sequence in triangular form
T := (n, k) -> n! * 2^(2*k-n) * binomial(n, k)*binomial(2*n-2*k, n-k):
for n from 0 to 8 do seq(T(n, k), k = 0..n) od;

Formula

T(n, k) = n! * 2^(2*k-n) * binomial(n, k) * binomial(2*n-2*k, n-k).
k*T(n, k) = (2*n^2)*T(n-1, k-1) for k >= 1 with T(n, 0) = (2*n - 1)!! = A001147(n).
T(n, 1) = 2*A161120(n).
T(n, n) = 2^n * n! = A000165(n); T(n+1, n) = 2^n * n! * (n+1)^2 = A014479(n);
T(n+2, n) = 3 * 2^(n-1)*(n+2)!*binomial(n+2, 2) = 3 * A286725(n).
More generally, T(n+r, n) = (2*r - 1)!! * A286724(n+r, r).
E.g.f.: Sum_{k >= 0} (1/2^k)*binomial(2*k, k)*t^k/(1 - 2*t*x)^(k+1) = 1 + (1 + 2*x)*t + (3 + 8*x + 8*x^2)*t^2/2! + (15 + 54*x + 72*x^2 + 48*x^3)*t^3/3! + ....
n-th row polynomial R(n, x) = (-2)^n*(x + x^2)^(n+1/2)*(d/dx)^n (1/sqrt(x + x^2)).
Recurrence for row polynomials:
R(n+1, x) = (2*x + 1)*(2*n + 1)*R(n, x) - 4*x*(x + 1)*n^2*R(n-1, x), with R(0, x) = 1.
R'(n, x) = 2*n^2 * R(n-1, x) for n >= 1.
Functional equation: R(n, -1 - x) = (-1)^n * R(n, x).
Conjecture: the zeros of the polynomial R(n, -x) lie on the vertical line Re(x) = 1/2 in the complex plane.
(-1)^n * x^n * R(n, (- 1 - x)/x) equals the n-th row polynomial of A123516.
(1 - x)^n * R(n, x/(1 - x)) equals the n-th row polynomial of A059366.
Let D denote the operator (1/x)*d/dx. Then D^(n+1)( arcsinh(x) ) = (-1)^n*R(n, x^2)/(x*sqrt(1 + x^2))^(2*n+1).
R(n, 1/2) = A331817(n); R(n, -1/2) = A177145(n+1);
(2^n) * R(n, 1/4) = A098461(n).
Alternating row sums R(n, -1) = (-1)^n * A001147(n).
Showing 1-4 of 4 results.

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