A082033 -id:A082033 - cp's OEIS frontend

A082032 Expansion of e.g.f.: exp(2x)/(1-2x).

1, 4, 20, 128, 1040, 10432, 125248, 1753600, 28057856, 505041920, 10100839424, 222218469376, 5333243269120, 138664325005312, 3882601100165120, 116478033004986368, 3727297056159629312, 126728099909427527680, 4562211596739391258624, 173364040676096868352000, 6934561627043874735128576

Offset: 0

Paul Barry, Apr 02 2003

Binomial transform of A010844. a(n) = b such that Integral_{x=0..1} (2*x)^n*exp(-x) dx = c - b*exp(-1). - Francesco Daddi, Jul 31 2011

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A000522, A010844, A082033.

With[{nn=30},CoefficientList[Series[Exp[2x]/(1-2x),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 02 2021 *)

my(x='x + O('x^25)); Vec(serlaplace(exp(2*x)/(1-2*x))) \\ Michel Marcus, Jan 27 2019

E.g.f.: exp(2*x)/(1-2*x)
a(n) = 2^n*A000522(n). - Vladeta Jovovic, Oct 29 2003
a(n) = 2n*a(n)+2^n, n>0, a(0)=1. - Paul Barry, Aug 26 2004
a(n) +2*(-n-1)*a(n-1) +4*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012
G.f.: 1/Q(0), where Q(k)= 1 - 2*x - 2*x*(k+1)/(1-2*x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
G.f.: 1/Q(0), where Q(k) = 1 - 4*x*(k+1) - 4*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
a(n) = 2^n*hypergeometric_U(1,n+2,1). - Peter Luschny, Nov 26 2014

More terms from Michel Marcus, Jan 27 2019

A320582 Number T(n,k) of permutations p of [n] such that |{ j : |p(j)-j| = 1 }| = k; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 2, 0, 4, 0, 5, 6, 10, 2, 1, 21, 36, 42, 12, 9, 0, 117, 226, 219, 104, 47, 6, 1, 792, 1568, 1472, 800, 328, 64, 16, 0, 6205, 12360, 11596, 6652, 2658, 688, 148, 12, 1, 55005, 109760, 103600, 60840, 24770, 7120, 1560, 200, 25, 0, 543597, 1085560, 1030649, 614420, 255830, 77732, 17750, 2876, 365, 20, 1

Offset: 0

Alois P. Heinz, Jan 23 2019

			T(4,0) = 5: 1234, 1432, 3214, 3412, 4231.
T(4,1) = 6: 2431, 3241, 3421, 4132, 4213, 4312.
T(4,2) = 10: 1243, 1324, 1342, 1423, 2134, 2314, 2413, 3124, 3142, 4321.
T(4,3) = 2: 2341, 4123.
T(4,4) = 1: 2143.
Triangle T(n,k) begins:
      1;
      1,      0;
      1,      0,      1;
      2,      0,      4,     0;
      5,      6,     10,     2,     1;
     21,     36,     42,    12,     9,    0;
    117,    226,    219,   104,    47,    6,    1;
    792,   1568,   1472,   800,   328,   64,   16,   0;
   6205,  12360,  11596,  6652,  2658,  688,  148,  12,  1;
  55005, 109760, 103600, 60840, 24770, 7120, 1560, 200, 25,  0;
  ...

Alois P. Heinz, Rows n = 0..24, flattened

Column k=0 gives A078480.
Row sums give A000142.
Main diagonal gives A059841.
Cf. A008290, A008291, A052582, A082033, A323671.

b:= proc(s) option remember; expand((n-> `if`(n=0, 1, add(
     `if`(abs(n-j)=1, x, 1)*b(s minus {j}), j=s)))(nops(s)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b({$1..n})):
seq(T(n), n=0..12);

b[s_] := b[s] = Expand[With[{n = Length[s]}, If[n==0, 1, Sum[
     If[Abs[n-j]==1, x, 1]*b[s~Complement~{j}], {j, s}]]]];
T[n_] := PadRight[CoefficientList[b[Range[n]], x], n+1];
T /@ Range[0, 12] // Flatten (* Jean-François Alcover, Feb 09 2021, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A052582(n-1) for n > 0.

Sum_{k=0..n} (k+1) * T(n,k) = A082033(n-1) for n > 0.

A082037 A square array of linear-factorial numbers, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 6, 1, 4, 10, 24, 24, 1, 5, 14, 42, 120, 120, 1, 6, 18, 60, 216, 720, 720, 1, 7, 22, 78, 312, 1320, 5040, 5040, 1, 8, 26, 96, 408, 1920, 9360, 40320, 40320, 1, 9, 30, 114, 504, 2520, 13680, 75600, 362880, 362880

Offset: 0

Paul Barry, Apr 02 2003

Rows include A000142(n), A000142(n+1), A007680, A082033, A082034. Columns include 2!*A005408(n),3!*A016777(n),4!*A016813(n),5!*A016861. Main diagonal is A082042.

			Rows begin
1 1 2 6 ....
1 2 6 24 ...
1 3 10 42 ...
1 4 14 60 ...
1 5 18 78 ...

Cf. A077038.

Square array defined by T(n, k)=(kn+1)n!

A082034 a(n) = (4n + 1)n!.

Original entry on oeis.org

1, 5, 18, 78, 408, 2520, 18000, 146160, 1330560, 13426560, 148780800, 1796256000, 23471078400, 330032102400, 4969162598400, 79768136448000, 1359981342720000, 24542432538624000, 467373280518144000, 9366672731480064000

Offset: 0

Paul Barry, Apr 02 2003

A row of the array A082037.

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A082033, A007680.

[(4*n+1)*Factorial(n) : n in [0..20]]; // Vincenzo Librandi, Sep 22 2011

Table[(4n+1)n!,{n,0,20}] (* Harvey P. Dale, Mar 02 2022 *)

a(n) = A016813(n)*n!.
(-4*n+3)*a(n) + n*(4*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 07 2014
4*a(n) + (-4*n-7)*a(n-1) + 3*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 07 2014

cp's OEIS Frontend

A082032 Expansion of e.g.f.: exp(2*x)/(1-2*x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A320582 Number T(n,k) of permutations p of [n] such that |{ j : |p(j)-j| = 1 }| = k; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A082037 A square array of linear-factorial numbers, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A082034 a(n) = (4*n + 1)*n!.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A082032 Expansion of e.g.f.: exp(2x)/(1-2x).

A082034 a(n) = (4n + 1)n!.