A181783 -id:A181783 - cp's OEIS frontend

A185106 Column 4 of A181783.

1, 7, 63, 709, 9709, 157971, 2993467, 64976353, 1593358809, 43632348319, 1321213523191, 43869502390077, 1585770335098693, 62013234471100459, 2609265444024424179, 117558236422872707161, 5647316731308685308337, 288166881285968665526583, 15566545814457889774570159, 887503412305357492886020789

Offset: 0

Richard Choulet, Dec 26 2012

A181783 is written as follows:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 4, 7, 11, 16, ...
1, 1, 5, 21, 63, 151, 311, ...
1, 1, 16, 142, 709, 2521, ...
1, 1, 65, 1201, 9709, ...
A000522 and A053482 are respectively the columns number 2 and 3 of this array. Our sequence gives the column number 4 (the fifth).

G. C. Greubel, Table of n, a(n) for n = 0..375

Cf. A181783, A000522, A053482.

a(0,1):=1:for p from 2 to 15 do for n from 0 to 20 do a(n,0):=1 :a(n,p):=n!*sum('1/(n-m)!*sum('k^(p-2)*(-1)^(p-1-k)*k^m/((k-1)!*(p-1-k)!)','k'=1..(p-1))','m'=0..n):od:seq(a(n,p),n=0..20):od;

CoefficientList[Series[E^x/((1-x)*(1-2x)*(1-3x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *)

x='x+O('x^50); Vec(serlaplace(exp(x)/((1-x)*(1-2*x)*(1-3*x)))) \\ G. C. Greubel, Jun 22 2017

Recurrence relation: a(n)= 6*n*a(n-1) -11*n*(n-1)*a(n-2) +6*n*(n-1)*(n-2)*a(n-3) +1 (or following A053482 for a linear homogeneous recurrence) a(n)= (6n+1)*a(n-1) -(11n+6)*(n-1)*a(n-2) +(6n+11)*(n-1)*(n-2)*a(n-3) -6*(n-1)*(n-2)*(n-3)*a(n-4).
E.g.f: exp(z)/((1-z)*(1-2*z)*(1-3*z)), as explained in A181783.
With p=4, a(n)=a(n,p)=n!*sum('1/(n-m)!*sum('k^(p-2)*(-1)^(p-1-k)*k^m/((k-1)!*(p-1-k)!)','k'=1..(p-1))','m'=0..n)
a(n) ~ n! * exp(1/3)*3^(n+2)/2. - Vaclav Kotesovec, Oct 02 2013

A053482 Binomial transform of A029767.

Original entry on oeis.org

1, 4, 21, 142, 1201, 12336, 149989, 2113546, 33926337, 611660476, 12243073621, 269456124774, 6468249055921, 168191402251432, 4709596238204901, 141291441773619106, 4521383010795364609, 153727989225714801396, 5534225015581836134677

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

This is the column k=3 of an array T(n,k) = A181783(n,k) defined by T(n,0)=T(0,k)=1 and T(n,k) = n*(k-1)*T(n-1,k) +T(n,k-1), which starts
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
   1,   1,   2,   4,   7,  11,  16,  22,  29,  37,  46,...
   1,   1,   5,  21,  63, 151, 311, 575, 981,1573,2401,...
   1,   1,  16, 142, 709,2521,7186,17536,38137,75889,140716,...
   1,   1,  65,1201,9709,50045,193765,614629,1682465,4110913,9176689,...
Column k=2 is A000522. The e.g.f. for column k is E_k(z) = E_(k-1)(z)/[1-(k-1)] = exp(z)/prod_{j=1..k-1} (1-j*z). - Richard Choulet, Dec 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200

CoefficientList[Series[E^x/(1-3*x+2*x^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *)

E.g.f.: exp(x)*(2/(1-2x)-1/(1-x))=exp(x)/(1-3x+2x^2); a(n)=sum{k=0..n, C(n,k)*k!*(2^(k+1)-1)}; a(n)=n!*sum{k=0..n, (2^(n-k+1)-1)/k!}; a(n)=int(x^n*(exp((1-x)/2)-exp(1-x)),x,1,infty); a(n)=2*A010844(n)-A000522(n); - Paul Barry, Jan 28 2008
Conjecture: a(n) -(3*n+1)*a(n-1) +(2*n+3)*(n-1)*a(n-2) -2*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 29 2012
a(n) = 3*n*a(n-2)-2*n*(n-1)*a(n-2)+1, derived from the array defined in the comment, which proves the previous conjecture. - Richard Choulet, Dec 17 2012
a(n) ~ n! * 2^(n+1)*exp(1/2). - Vaclav Kotesovec, Oct 02 2013

A371766 Triangle read by rows: T(n, k) = A371898(n, k) / A371767(n, k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 16, 21, 7, 1, 1, 65, 142, 63, 11, 1, 1, 326, 1201, 709, 151, 16, 1, 1, 1957, 12336, 9709, 2521, 311, 22, 1, 1, 13700, 149989, 157971, 50045, 7186, 575, 29, 1, 1, 109601, 2113546, 2993467, 1158871, 193765, 17536, 981, 37, 1

Offset: 0

Peter Luschny, Apr 14 2024

			Triangle starts:
  [0] 1;
  [1] 1,     1;
  [2] 1,     2,      1;
  [3] 1,     5,      4,      1;
  [4] 1,    16,     21,      7,     1;
  [5] 1,    65,    142,     63,    11,    1;
  [6] 1,   326,   1201,    709,   151,   16,   1;
  [7] 1,  1957,  12336,   9709,  2521,  311,  22,  1;
  [8] 1, 13700, 149989, 157971, 50045, 7186, 575, 29, 1;

Antidiagonally read subtriangle of A181783.

Cf. A371898, A371767.

A371766 := (n, k) -> local j; add((-1)^(k-j)*binomial(k, j)*hypergeom([1, -n],
[], -j), j = 0..k)/((k! * n!)/(n - k)!):
seq(print(seq(simplify(A371766(n, k)), k = 0..n)), n = 0..8);

cp's OEIS Frontend

A185106 Column 4 of A181783.

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A053482 Binomial transform of A029767.

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A371766 Triangle read by rows: T(n, k) = A371898(n, k) / A371767(n, k).

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Maple