A000457 -id:A000457 - cp's OEIS frontend

A162974 Triangle read by rows: T(n,k) is the number of derangements of {1,2,...,n} having k cycles of length 2 (0 <= k <= floor(n/2)).

1, 0, 0, 1, 2, 0, 6, 0, 3, 24, 20, 0, 160, 90, 0, 15, 1140, 504, 210, 0, 8988, 4480, 1260, 0, 105, 80864, 41040, 9072, 2520, 0, 809856, 404460, 100800, 18900, 0, 945, 8907480, 4447520, 1128600, 166320, 34650, 0, 106877320, 53450496, 13347180, 2217600

Offset: 0

Emeric Deutsch, Jul 22 2009

Row n has 1 + floor(n/2) entries.
Sum of entries in row n = A000166(n) (the derangement numbers).
T(n,0) = A038205(n).
Sum_{k>=0} k*T(n,k) = A000387(n).

			T(4,2)=3 because we have (12)(34), (13)(24), and (14)(23).
Triangle starts:
    1;
    0;
    0,  1;
    2,  0;
    6,  0,  3;
   24, 20,  0;
  160, 90,  0, 15;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000166, A000387, A038205, A114320.
T(2n,n) gives A001147.
T(2n+3,n) gives A000906(n) = 2*A000457(n).

G := exp((1/2)*z*(t*z-z-2))/(1-z): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 13 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: for n from 0 to 13 do seq(coeff(P[n], t, j), j = 0 .. floor((1/2)*n)) end do;
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add((j-1)!*
      `if`(j=2, x, 1)*b(n-j)*binomial(n-1, j-1), j=2..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..14);  # Alois P. Heinz, Jan 27 2022

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[(j - 1)!*If[j == 2, x, 1]*b[n - j]*Binomial[n - 1, j - 1], {j, 2, n}]]];
T[n_] := With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Sep 17 2024, after Alois P. Heinz *)

E.g.f.: G(t,z) = exp(z(tz-z-2)/2)/(1-z).

A259877 If n is even then a(n) = n!/( 2^(n/2)(n/2)! ), otherwise a(n) = n!/( 32^((n-1)/2)*((n-3)/2)! ).

Original entry on oeis.org

1, 1, 3, 10, 15, 105, 105, 1260, 945, 17325, 10395, 270270, 135135, 4729725, 2027025, 91891800, 34459425, 1964187225, 654729075, 45831035250, 13749310575, 1159525191825, 316234143225, 31623414322500, 7905853580625, 924984868933125, 213458046676875, 28887988983603750, 6190283353629375

Offset: 2

N. J. A. Sloane, Jul 09 2015

nonn

Chai Wah Wu, Table of n, a(n) for n = 2..501
D. L. Andrews, Letter to N. J. A. Sloane, Apr 10 1978.
D. L. Andrews and T. Thirunamachandran, On three-dimensional rotational averages, J. Chem. Phys., 67 (1977), 5026-5033. See N_n.
D. L. Andrews and T. Thirunamachandran, On three-dimensional rotational averages, J. Chem. Phys., 67 (1977), 5026-5033. [Annotated scanned copy]

A001147 alternating with A000457. Interlaced diagonal of A008299.

f:=proc(n) if n mod 2 = 0 then
n!/(2^(n/2)*(n/2)!) else
n!/( 3*2^((n-1)/2)*((n-3)/2)! ); fi; end;
[seq(f(n),n=2..30)];

Table[(n!/6)*2^(-n/2)*(((2^(1/2)*(1-(-1)^n))/(n/2-3/2)!)+3*(1+(-1)^n)/(n/2)!), {n, 2, 30}] (* Wesley Ivan Hurt, Jul 10 2015 *)

main(size)={v=vector(size);for(n=2, size+1, if(n%2==0, v[n-1]=n!/(2^(n/2)*(n/2)!), v[n-1]=n!/( 3*2^((n-1)/2)*((n-3)/2)!))); return(v);} /* Anders Hellström, Jul 10 2015 */

from _future_ import division
A259877_list, a = [1], 1
for n in range(2,10**2):
    a = 6*a//(n-1) if n % 2 else a*n*(n+1)//6
    A259877_list.append(a) # Chai Wah Wu, Jul 15 2015

a(n) = (n!/6)*2^(-n/2)*(((2^(1/2)*(1-(-1)^n))/(n/2-3/2)!)+3*(1+(-1)^n)/(n/2)!). - Wesley Ivan Hurt, Jul 10 2015
a(n+1) = a(n)*n*(n+1)/6 if n is even, a(n+1) = 6*a(n)/(n-1) if n is odd. - Chai Wah Wu, Jul 15 2015
a(2*n) = A001147(n), a(2*n+1) = A000457(n-1). - Yuchun Ji, Nov 02 2020

A275521 Number of (n+floor(n/2))-block bicoverings of an n-set.

Original entry on oeis.org

1, 0, 1, 4, 3, 40, 15, 420, 105, 5040, 945, 69300, 10395, 1081080, 135135, 18918900, 2027025, 367567200, 34459425, 7856748900, 654729075, 183324141000, 13749310575, 4638100767300, 316234143225, 126493657290000, 7905853580625, 3699939475732500, 213458046676875

Offset: 0

Alois P. Heinz, Jul 31 2016

nonn

There are no bicoverings of an n-set with more than n+floor(n/2) blocks.

			a(2) = 1: 1|12|2.
a(3) = 4: 1|12|23|3, 1|13|2|23, 1|123|2|3, 12|13|2|3.
a(4) = 3: 1|12|2|3|34|4, 1|13|2|24|3|4, 1|14|2|23|3|4.

Alois P. Heinz, Table of n, a(n) for n = 0..808

Right border of triangle A059443.

Bisections give: A001147, 4*A000457(n-1) (for n>0).

a:= proc(n) option remember; `if`(n<5, [1, 0, 1, 4, 3]
       [n+1], ((8*n-41)*a(n-1) +(6*n^2-12*n-12)*a(n-2)
       -(n-2)*(8*n-17)*a(n-3)) / (6*n-24))
    end:
seq(a(n), n=0..30);

a(n) = A059443(n,n+floor(n/2)).

A327411 a(n) = multinomial(2*n+3; 3, 2, 2, ..., 2) (n times '2').

Original entry on oeis.org

1, 10, 210, 7560, 415800, 32432400, 3405402000, 463134672000, 79196028912000, 16631166071520000, 4207685016094560000, 1262305504828368000000, 443069232194757168000000, 179886108271071410208000000, 83647040346048205746720000000, 44165637302713452634268160000000

Offset: 0

Peter Luschny, Sep 07 2019

nonn

Cf. A000457, A327412.

a:= n-> combinat[multinomial](2*n+3, 3, 2$n):
seq(a(n), n=0..17);  # Alois P. Heinz, Sep 07 2019

multinomial[n_, k_List] := n!/Times @@ (k!);
a[n_] := multinomial[2n+3, Join[{3}, Table[2, {n}]]];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Apr 19 2025 *)

def a(n):
    return multinomial([3] + [2] * n)
[a(n) for n in range(20)]

a(n) = 2^(-n-1)*Gamma(2*n + 4)/3.
a(n) = (2*(n-1)^2 + 9*n + 1)*a(n-1)  for n > 0.
a(n) / n! = A000457(n).

A334824 Triangle, read by rows, of Lambert's numerator polynomials related to convergents of tan(x).

Original entry on oeis.org

1, 3, 0, 15, 0, -1, 105, 0, -10, 0, 945, 0, -105, 0, 1, 10395, 0, -1260, 0, 21, 0, 135135, 0, -17325, 0, 378, 0, -1, 2027025, 0, -270270, 0, 6930, 0, -36, 0, 34459425, 0, -4729725, 0, 135135, 0, -990, 0, 1, 654729075, 0, -91891800, 0, 2837835, 0, -25740, 0, 55, 0, 13749310575, 0, -1964187225, 0, 64324260, 0, -675675, 0, 2145, 0, -1

Offset: 0

G. C. Greubel, May 13 2020, following a suggestion from Michel Marcus

Lambert's denominator polynomials related to convergents of tan(x), f(n, x), are given in A334823.

			Polynomials:
g(0, x) = 1;
g(1, x) = 3*x;
g(2, x) = 15*x^2 - 1;
g(3, x) = 105*x^3 - 10*x;
g(4, x) = 945*x^4 - 105*x^2 + 1;
g(5, x) = 10395*x^5 - 1260*x^3 + 21*x;
g(6, x) = 135135*x^6 - 17325*x^4 + 378*x^2 - 1;
g(7, x) = 2027025*x^7 - 270270*x^5 + 6930*x^3 - 36*x.
Triangle of coefficients begins as:
        1;
        3, 0;
       15, 0,      -1;
      105, 0,     -10, 0;
      945, 0,    -105, 0,    1;
    10395, 0,   -1260, 0,   21, 0;
   135135, 0,  -17325, 0,  378, 0,  -1;
  2027025, 0, -270270, 0, 6930, 0, -36, 0.

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
J.-H. Lambert, Mémoire sur quelques propriétés remarquables des quantités transcendantes et logarithmiques (Memoir on some properties that can be traced from circular transcendent and logarithmic quantities), Histoire de l’Académie royale des sciences et belles-lettres (1761), Berlin. See also.

Cf. A001497, A001498, A053983, A053984, A053987, A053988, A094675, A334823.

Columns k: A001147 (k=0), A000457 (k=2), A001881 (k=4), A130563 (k=6).

C := ComplexField();
T:= func< n, k| Round( i^k*Factorial(2*n-k+1)*(1+(-1)^k)/(2^(n-k+1)*Factorial(k+1)*Factorial(n-k)) ) >;
[T(n,k): k in [0..n], n in [0..10]];

Maple

T:= (n, k) -> I^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!); seq(seq(T(n, k), k = 0..n), n = 0..10);

Mathematica

(* First program *) y[n_, x_]:= Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n -1/2, 1/x]; g[n_, k_]:= Coefficient[((-I)^n/2)*(y[n+1, I*x] + (-1)^n*y[n+1, -I*x]), x, k]; Table[g[n, k], {n,0,10}, {k,n,0,-1}]//Flatten (* Second program *) Table[I^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!), {n,0,10}, {k,0,n}]//Flatten

Sage

[[ i^k*factorial(2*n-k+1)*(1+(-1)^k)/(2^(n-k+1)*factorial(k+1)*factorial(n-k)) for k in (0..n)] for n in (0..10)]

Formula

Equals the coefficients of the polynomials, g(n, x), defined by: (Start)

g(n, x) = Sum_{k=0..floor(n/2)} ((-1)^k*(2*n-2*k+1)!/((2*k+1)!*(n-2*k)!))*(x/2)^(n-2*k).

g(n, x) = ((2*n+1)!/n!)*(x/2)^n*Hypergeometric2F3(-n/2, (1-n)/2; 3/2, -n, -n-1/2; -1/x^2).

g(n, x) = ((-i)^n/2)*(y(n+1, i*x) + (-1)^n*y(n+1, -i*x)), where y(n, x) are the Bessel Polynomials.

g(n, x) = (2*n-1)*x*g(n-1, x) - g(n-2, x).

E.g.f. of g(n, x): sin((1 - sqrt(1-2*x*t))/2)/sqrt(1-2*x*t).

g(n, 1) = (-1)^n*g(n, -1) = A053984(n) = (-1)^n*A053983(-n-1) = (-1)^n*f(-n-1, 1).

g(n, 2) = (-1)^n*g(n, -2) = A053987(n+1). (End)

As a number triangle:

T(n, k) = i^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!), where i = sqrt(-1).

T(n, 0) = A001147(n+1).

A112000 One half of third column (k=2) of triangle A111999.

Original entry on oeis.org

-3, 65, -1190, 22050, -433125, 9144135, -208107900, 5099994900, -134219460375, 3781060408125, -113633468798850, 3631422078033750, -123022987568105625, 4405418319999571875, -166312279434175875000, 6602853358582065585000, -275059081486584416896875

Offset: 0

Wolfdieter Lang, Sep 12 2005

Cf. Second (k=1) column: A000906(n+2)*(-1)^n = 2*A000457(n+2)*(-1)^n, n>=0.

a(n)=A111999(n+3, 2)/2, n>=0.

Conjecture: +n*(4*n+5)*a(n) +(2*n+3)*(n+2)*(4*n+9)*a(n-1)=0. - R. J. Mathar, Jul 09 2017

A380281 Triangle T(n, k) read by rows: T(n, k) = 2^nbinomial(2n + 1, 2k + 1) Pochhammer(1/2, n - k) * Pochhammer(1/2, k).

Original entry on oeis.org

1, 3, 1, 15, 10, 3, 105, 105, 63, 15, 945, 1260, 1134, 540, 105, 10395, 17325, 20790, 14850, 5775, 945, 135135, 270270, 405405, 386100, 225225, 73710, 10395, 2027025, 4729725, 8513505, 10135125, 7882875, 3869775, 1091475, 135135, 34459425, 91891800, 192972780, 275675400, 268017750, 175429800, 74220300, 18378360, 2027025

Offset: 0

Thomas Scheuerle, Jan 18 2025

			Triangle begins:
n\k      0 |      1 |      2 |      3 |      4 |       5 |      6 |     7 |
[0]       1,
[1]       3,       1
[2]      15,      10,       3
[3]     105,     105,      63,       15
[4]     945,    1260,    1134,      540,     105
[5]   10395,   17325,   20790,    14850,    5775,     945
[6]  135135,  270270,  405405,   386100,  225225,   73710,   10395
[7] 2027025, 4729725, 8513505, 10135125, 7882875, 3869775, 1091475, 135135

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
C. Nicholson, The probability integral for two variables, Biometrika 33 (1943), 59-72.
Eric Weisstein's World of Mathematics, Owen T-Function.

T(n, 1) = A001147(n+1), T(n, 2) = A000457(n-1), T(n, 3) = A001881(n+3)*3, T(n, n) = A001147(n).

Cf. A076729, (conj. row sums), A103327, A173424.

T := (n,k) -> 2^n*binomial(2*n + 1, 2*k + 1)*pochhammer(1/2, n - k)*pochhammer(1/2, k):
for n from 0 to 7 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Jan 21 2025

A380281[n_, k_] := (2*n - 1)!!*Binomial[n, k]*Binomial[2*n + 1, 2*k + 1]/Binomial[2*n, 2*k];
Table[A380281[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 22 2025 *)

T(n, k) = Vec(O(x^(1+n))+(1+x)^(n+1)*hypergeom([1,1/2+n+1],3/2,-x)*(2*(n+1))!/(2^(n+1)*(n+1)!))[1+k]

doublefact(n) = prod(i=0, (n-1)\2, n - 2*i )
T(n, k) = doublefact(2*n-1) * binomial(n, k) * binomial(2*n+1, 2*k+1) / binomial(2*n, 2*k)

rf = rising_factorial
def T(n, k): return 2^n*binomial(2*n+1, 2*k+1)*rf(1/2, n-k)*rf(1/2, k)
for n in range(9): print([T(n, k) for k in range(n+1)])  # Peter Luschny, Jan 21 2025

Coefficients for the series representation of Owen's T-function Ot(x, m) = atan(m)/(2*Pi) + Sum_{s>=0} (-1)^(s+1)*m*(Sum_{r=0..s} T(s, r)*m^(2*s))*x^(2+2*s)/(2*Pi*(2+2*s)!).
Ot(x, m) - atan(m)/(2*Pi) = -V(x, x*m), where V is Nicholson's V-function. V(h, q) = Integral_{x=0..h} Integral_{y=0..q*x/h} phi(x)*phi(y) dydx, where phi(x) is the standard normal density exp(-x^2/2)/sqrt(2*Pi).
G.f. of row n: ((1 + x)^(n+1)*Hypergeometric2F1[1, 1/2 + n + 1, 3/2, -x]*(2*(n+1))!)/(2^(n+1)*(n+1)!).
T(n, k) = A103327(n, k)*A173424(n, k).
T(n, k) = (2*n-1)!! * binomial(n, k) * binomial(2*n+1, 2*k+1) / binomial(2*n, 2*k).
Conjecture: Row sums are A076729.

cp's OEIS Frontend

A162974 Triangle read by rows: T(n,k) is the number of derangements of {1,2,...,n} having k cycles of length 2 (0 <= k <= floor(n/2)).

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Maple

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A259877 If n is even then a(n) = n!/( 2^(n/2)*(n/2)! ), otherwise a(n) = n!/( 3*2^((n-1)/2)*((n-3)/2)! ).

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Python

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A275521 Number of (n+floor(n/2))-block bicoverings of an n-set.

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Maple

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A327411 a(n) = multinomial(2*n+3; 3, 2, 2, ..., 2) (n times '2').

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SageMath

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A334824 Triangle, read by rows, of Lambert's numerator polynomials related to convergents of tan(x).

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Magma

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Mathematica

Sage

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A112000 One half of third column (k=2) of triangle A111999.

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A380281 Triangle T(n, k) read by rows: T(n, k) = 2^n*binomial(2*n + 1, 2*k + 1) * Pochhammer(1/2, n - k) * Pochhammer(1/2, k).

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Maple

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PARI

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A259877 If n is even then a(n) = n!/( 2^(n/2)(n/2)! ), otherwise a(n) = n!/( 32^((n-1)/2)*((n-3)/2)! ).

A380281 Triangle T(n, k) read by rows: T(n, k) = 2^nbinomial(2n + 1, 2k + 1) Pochhammer(1/2, n - k) * Pochhammer(1/2, k).