A255806 -id:A255806 - cp's OEIS frontend

A082579 Expansion of e.g.f.: exp( x/(1-x)^2 ).

1, 1, 5, 31, 241, 2261, 24781, 309835, 4342241, 67308841, 1141960501, 21026890391, 417264626065, 8871853115581, 201100863674621, 4838817223845571, 123128720142540481, 3302478863343928145, 93091427773284348901, 2750635764338982054031, 84994418675445218025521

Offset: 0

Emanuele Munarini, May 07 2003

Old name: A binomial sum.
a(n) is the number of ways that n people can form any number of lines and then designate one person in each line.  Equivalently, number of ways to linearly arrange the elements in each block of a set partition, then underline one element in each block summed over all set partitions of {1,2,...,n}. a(2) = 5: [1'][2'], [1',2], [1,2'], [2',1], [2,1']. - Geoffrey Critzer, Nov 04 2012
It appears that the sequence taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k would be periodic with period dividing k. - Peter Bala, Nov 14 2017
The above conjecture is true - see the Bala link. - Peter Bala, Jan 20 2018

Seiichi Manyama, Table of n, a(n) for n = 0..433
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A000262, A052897, A255806, A255807, A255819.

A082579:= func< n | n eq 0 select 1 else (&+[Factorial(n)*Binomial(n+k-1, n-k)/Factorial(k): k in [1..n]]) >;
[A082579(n): n in [0..25]]; // G. C. Greubel, Feb 23 2021

nn=20;Range[0,nn]!CoefficientList[Series[Exp[ x/(1-x)^2],{x,0,nn}],x]  (* Geoffrey Critzer, Nov 04 2012 *)
nn = 20; Range[0, nn]! * CoefficientList[Series[Product[Exp[k*x^k], {k, 1, nn}], {x, 0, nn}], x] (* Vaclav Kotesovec, Mar 21 2016 *)
Table[If[n==0, 1, n*n!*HypergeometricPFQ[{1-n, n+1}, {3/2, 2}, -1/4]], {n, 0, 25}] (* G. C. Greubel, Feb 23 2021 *)

a(n):=n!*sum(binomial(n+k-1,2*k-1)/k!,k,1,n); /* Vladimir Kruchinin, Apr 21 2011 */

my(x='x+O('x^33));
Vec(serlaplace(exp( x/(1-x)^2 )))
/* Joerg Arndt, Sep 14 2012 */

[1 if n==0 else factorial(n)*sum( binomial(n+k-1, n-k)/factorial(k) for k in (1..n)) for n in (0..25)] # G. C. Greubel, Feb 23 2021

a(n) = n!*Sum_{k=0..n} binomial(n+k-1, 2*k-1)/k!.
Recurrence: a(n+3) - (3*n+7)*a(n+2) + (n+2)*(3*n+2)*a(n+1) - (n+2)*(n+1)*n*a(n) = 0.
E.g.f.: exp( x/( 1 - x )^2 ).
Special values of the hypergeometric function 2F2: a(n)=n!*n*hypergeom([n+1, -n+1], [3/2, 2], -1/4), n >= 1. - Karol A. Penson, Jan 29 2004
a(n) ~ 2^(1/6)*n^(n-1/6)*exp(-1/12 + 3*(n/2)^(2/3) - n)/sqrt(3). - Vaclav Kotesovec, Jun 26 2013
E.g.f.: E(0)/2, where E(k) = 1 + 1/( 1 - x/(x + (1-x)^2*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
E.g.f.: exp(Sum_{k>=1} k*x^k). - Vaclav Kotesovec, Mar 07 2015
a(n) = n!*y(n), with y(0) = 1, y(n) = (Sum_{k=0..n-1} (n-k)^2*y(k))/n. - Benedict W. J. Irwin, Jun 02 2016
E.g.f.: Product_{k>=1} 1/(1 - x^k)^(J_2(k)/k), where J_2() is the Jordan function (A007434). - Ilya Gutkovskiy, May 25 2019
a(n) = n*n!*Hypergeometric2F2([1-n, n+1], [3/2, 2], -1/4) with a(0) = 1. - G. C. Greubel, Feb 23 2021

A052897 Expansion of e.g.f.: exp(2*x/(1-x)).

Original entry on oeis.org

1, 2, 8, 44, 304, 2512, 24064, 261536, 3173888, 42483968, 621159424, 9841950208, 167879268352, 3065723549696, 59651093528576, 1231571119812608, 26883546193002496, 618463501807058944

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 873
N. J. A. Sloane, Transforms
Index entries for sequences related to Laguerre polynomials

Row sums of A059110.

Cf. A000262, A025168, A255806, A317364.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(2*x/(1 - x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2018

[Factorial(n)*Evaluate(LaguerrePolynomial(n, -1), -2): n in [0..25]]; // G. C. Greubel, Feb 23 2021

L := proc(n,a,x) if n=0 then 1 elif n=1 then a+1-x else (2*n+a-1-x)/n*L(n-1,a,x) - (n+a-1)/n*L(n-2,a,x) fi end: A052897 := n -> n!*L(n,-1,-2): seq(A052897(n),n=0..17); # Peter Luschny, Nov 20 2011
spec := [S,{B=Set(C),C=Sequence(Z,1 <= card),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Range[0, 19]! CoefficientList[ Series[E^(2*x/(1 - x)), {x, 0, 19}], x] (* Zerinvary Lajos, Mar 21 2007 *)
Table[n!*LaguerreL[n, -1, -2], {n,0,30}] (* G. C. Greubel, Feb 23 2021 *)

a=Vec(exp(2*x/(1-x)));for(n=2,#a-1,a[n+1]*=n!);a \\ Charles R Greathouse IV, Nov 20 2011

[factorial(n)*gen_laguerre(n, -1, -2) for n in (0..25)] # G. C. Greubel, Feb 23 2021

E.g.f.: exp(2*x/(1-x)). - Vladeta Jovovic, Jan 04 2001
Recurrence: {a(0)=1, a(1)=2, (n^2+n)*a(n) + (-4-2*n)*a(n+1) + a(n+2)}.
LAH transform of A000079: a(n) = Sum_{k=0..n} 2^k*n!/k!*binomial(n-1, k-1). - Vladeta Jovovic, Oct 17 2003
a(n) = n!*L(n,-1,-2). - Karol A. Penson, Oct 16 2006 [Here L(n,a,x) is the n-th generalized Laguerre polynomial with parameter a, evaluated at x. L(n,a,x) is 1 if n=0, a+1-x if n=1 and otherwise (2*n+a-1-x)/n*L(n-1,a,x)-(n+a-1)/n*L(n-2,a,x). - Peter Luschny, Nov 20 2011]
a(n) ~ 2^(-1/4)*exp(2*sqrt(2*n)-n-1)*n^(n-1/4) * (1 + 7/(48*sqrt(2*n))). - Vaclav Kotesovec, Oct 09 2012, extended Dec 01 2021
E.g.f.: 1 + 2*x/((1-x)*T(0) - x), where T(k) = 4*k+1 + x^2/((4*k+3)*(1-x)^2 + x^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 30 2013
E.g.f.: exp(Sum_{k>=1} 2*x^k). - Vaclav Kotesovec, Mar 07 2015
a(n) = Sum_{k=0..n} binomial(n,k)*l(k)*l(n-k), where l(m) = A000262(m). - Emanuele Munarini, Aug 31 2017

New name using e.g.f., Vaclav Kotesovec, Feb 25 2014

A253286 Square array read by upward antidiagonals, A(n,k) = Sum_{j=0..n} (n-j)!C(n,n-j) C(n-1,n-j)*k^j, for n>=0 and k>=0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 13, 8, 3, 1, 0, 73, 44, 15, 4, 1, 0, 501, 304, 99, 24, 5, 1, 0, 4051, 2512, 801, 184, 35, 6, 1, 0, 37633, 24064, 7623, 1696, 305, 48, 7, 1, 0, 394353, 261536, 83079, 18144, 3145, 468, 63, 8, 1

Offset: 0

Peter Luschny, Mar 24 2015

			Square array starts, A(n,k):
      1,       1,       1,       1,      1,      1,      1, ...  A000012
      0,       1,       2,       3,      4,      5,      6, ...  A001477
      0,       3,       8,      15,     24,     35,     48, ...  A005563
      0,      13,      44,      99,    184,    305,    468, ...  A226514
      0,      73,     304,     801,   1696,   3145,   5328, ...
      0,     501,    2512,    7623,  18144,  37225,  68976, ...
      0,    4051,   24064,   83079, 220096, 495475, 997056, ...
A000007, A000262, A052897, A255806, ...
Triangle starts, T(n, k) = A(n-k, k):
  1;
  0,   1;
  0,   1,   1;
  0,   3,   2,  1;
  0,  13,   8,  3,  1;
  0,  73,  44, 15,  4, 1;
  0, 501, 304, 99, 24, 5, 1;

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Laguerre polynomials

Main diagonal gives A293145.

Cf. A000262, A001477, A005563, A052897, A226514, A255806, A256325.

[k eq n select 1 else k*Factorial(n-k-1)*Evaluate(LaguerrePolynomial(n-k-1, 1), -k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 23 2021

L := (n, k) -> (n-k)!*binomial(n,n-k)*binomial(n-1,n-k):
A := (n, k) -> add(L(n,j)*k^j, j=0..n):
# Alternatively:
# A := (n, k) -> `if`(n=0,1, simplify(k*n!*hypergeom([1-n],[2],-k))):
for n from 0 to 6 do lprint(seq(A(n,k), k=0..6)) od;

A253286[n_, k_]:= If[k==n, 1, k*(n-k-1)!*LaguerreL[n-k-1, 1, -k]];
Table[A253286[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 23 2021 *)

{T(n, k) = if(n==0, 1, n!*sum(j=1, n, k^j*binomial(n-1, j-1)/j!))} \\ Seiichi Manyama, Feb 03 2021

{T(n, k) = if(n<2, (k-1)*n+1, (2*n+k-2)*T(n-1, k)-(n-1)*(n-2)*T(n-2, k))} \\ Seiichi Manyama, Feb 03 2021

flatten([[1 if k==n else k*factorial(n-k-1)*gen_laguerre(n-k-1, 1, -k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 23 2021

A(n,k) = k*n!*hypergeom([1-n],[2],-k) for n>=1 and 1 for n=0.
Row sums of triangle, Sum_{k=0..n} A(n-k, k) = 1 + A256325(n).
From Seiichi Manyama, Feb 03 2021: (Start)
E.g.f. of column k: exp(k*x/(1-x)).
T(n,k) = (2*n+k-2) * T(n-1,k) - (n-1) * (n-2) * T(n-2, k) for n > 1. (End)
From G. C. Greubel, Feb 23 2021: (Start)
A(n, k) = k*(n-1)!*LaguerreL(n-1, 1, -k) with A(0, k) = 1.
T(n, k) = k*(n-k-1)!*LaguerreL(n-k-1, 1, -k) with T(n, n) = 1.
T(n, 2) = A052897(n) = A086915(n)/2.
Sum_{k=0..n} T(n, k) = 1 + Sum_{k=0..n-1} (n-k-1)*k!*LaguerreL(k, 1, k-n+1). (End)

A293145 a(n) = n! * [x^n] exp(n*x/(1 - x)).

Original entry on oeis.org

1, 1, 8, 99, 1696, 37225, 997056, 31535371, 1150303232, 47538819729, 2195314048000, 112032721984051, 6261138045038592, 380309520560089081, 24946892219825709056, 1757549042234670166875, 132356128415391650676736, 10610067001068927596601889, 902057202129607760380428288

Offset: 0

Ilya Gutkovskiy, Oct 01 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..356
Index entries for sequences related to Laguerre polynomials

Main diagonal of A253286.

Cf. A000262, A052857, A052897, A255806, A277372.

[n eq 0 select 1 else Factorial(n)*Evaluate(LaguerrePolynomial(n-1, 1), -n): n in [0..20]]; // G. C. Greubel, Feb 23 2021

Table[n! SeriesCoefficient[Exp[n x/(1 - x)], {x, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[Product[Exp[n x^k], {k, 1, n}], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[Sum[n^k n!/k! Binomial[n - 1, k - 1], {k, n}], {n, 1, 18}]]
Join[{1}, Table[n n! Hypergeometric1F1[1 - n, 2, -n], {n, 1, 18}]]
Table[If[n==0, 1, n!*LaguerreL[n-1, 1, -n]], {n, 0, 20}] (* G. C. Greubel, Feb 23 2021 *)

{a(n) = if(n==0, 1, n!*sum(k=1, n, n^k*binomial(n-1, k-1)/k!))} \\ Seiichi Manyama, Feb 03 2021

a(n) = if (n, n! * pollaguerre(n-1, 1, -n), 1); \\ Michel Marcus, Feb 23 2021

[1 if n==0 else factorial(n)*gen_laguerre(n-1, 1, -n) for n in (0..20)] # G. C. Greubel, Feb 23 2021

a(n) = n! * [x^n] Product_{k>=1} exp(n*x^k).
a(n) ~ exp(n/phi - n) * phi^(2*n) * n^n / 5^(1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 01 2017
a(n) = n! * Sum_{k=1..n} n^k * binomial(n-1,k-1)/k! for n > 0. - Seiichi Manyama, Feb 03 2021
a(n) = n! * LaguerreL(n-1, 1, -n) with a(0) = 1. - G. C. Greubel, Feb 23 2021

cp's OEIS Frontend

A082579 Expansion of e.g.f.: exp( x/(1-x)^2 ).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Sage

Formula

A052897 Expansion of e.g.f.: exp(2*x/(1-x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A253286 Square array read by upward antidiagonals, A(n,k) = Sum_{j=0..n} (n-j)!*C(n,n-j)* C(n-1,n-j)*k^j, for n>=0 and k>=0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A293145 a(n) = n! * [x^n] exp(n*x/(1 - x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

A253286 Square array read by upward antidiagonals, A(n,k) = Sum_{j=0..n} (n-j)!C(n,n-j) C(n-1,n-j)*k^j, for n>=0 and k>=0.