A052897 -id:A052897 - 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

A059110 Triangle T = A007318A271703; T(n,m)= Sum_{i=0..n} L'(n,i)binomial(i,m), m=0..n.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 13, 21, 9, 1, 73, 136, 78, 16, 1, 501, 1045, 730, 210, 25, 1, 4051, 9276, 7515, 2720, 465, 36, 1, 37633, 93289, 85071, 36575, 8015, 903, 49, 1, 394353, 1047376, 1053724, 519456, 137270, 20048, 1596, 64, 1, 4596553, 12975561

Offset: 0

Vladeta Jovovic, Jan 04 2001

L'(n,i) are unsigned Lah numbers (cf. A008297): L'(n,i)=n!/i!*binomial(n-1,i-1) for i >= 1, L'(0,0)=1, L'(n,0)=0 for n>0. T(n,0)=A000262(n); T(n,2)=A052852(n). Row sums A052897.
Exponential Riordan array [e^(x/(1-x)),x/(1-x)]. - Paul Barry, Apr 28 2007
From Wolfdieter Lang, Jun 22 2017: (Start)
The inverse matrix T^(-1) is exponential Riordan (aka Sheffer) (e^(-x), x/(1+x)): T^(-1)(n, m) = (-1)^(n-m)*A271705(n, m).
The a- and z-sequences of this Sheffer (aka exponential Riordan) matrix are a = [1,1,repeat(0)] and z(n) = (-1)^(n+1)*A028310(n)/A000027(n-1) with e.g.f. ((1+x)/x)*(1-exp(-x)). For a- and z-sequences see a W. Lang link under A006232 with references. (End)

			The triangle T = A007318*A271703 starts:
n\m       0        1        2       3       4      5     6    7  8 9 ...
0:        1
1:        1        1
2:        3        4        1
3:       13       21        9       1
4:       73      136       78      16       1
5:      501     1045      730     210      25      1
6:     4051     9276     7515    2720     465     36     1
7:    37633    93289    85071   36575    8015    903    49    1
8:   394353  1047376  1053724  519456  137270  20048  1596   64  1
9:  4596553 12975561 14196708 7836276 2404206 427518 44436 2628 81 1
... reformatted. - _Wolfdieter Lang_, Jun 22 2017
E.g.f. for T(n, 2) = 1/2!*(x/(1-x))^2*e^(x/(x-1)) = 1*x^2/2 + 9*x^3/3! + 78*x^4/4! + 730*x^5/5! + 7515*x^6/6 + ...
From _Wolfdieter Lang_, Jun 22 2017: (Start)
The z-sequence starts: [1, 1/2, -2/3, 3/4, -4/5, 5/6, -6/7, 7/8, -8/9, ...
T recurrence: T(3, 0) = 3*(1*T(2,0) + (1/2)*T(2, 1) + (-2/3)*T(2 ,1)) = 3*(3 + (1/2)*4 - (2/3)) = 13; T(3, 1) = 3*(T(2, 0)/1 + T(2, 1)) = 3*(3 + 4) = 21.
Meixner type recurrence for R(2, x): (D - D^2)*(3 + 4*x + x^2) = 4 + 2*x - 2 = 2*(1 + x), (D = d/dx).
General Sheffer recurrence for R(2, x): (1+x)*(1 + 2*D + D^2)*(1 + x) = (1+x)*(1 + x + 2) = 3 + 4*x + x^2. (End)

Muniru A Asiru, Rows n=0..50 of triangle, flattened
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A007318, A008297, A052852, A052897, A271703, A271705.

Concatenation([1],Flat(List([1..10],n->List([0..n],m->Sum([0..n],i-> Factorial(n)/Factorial(i)*Binomial(n-1,i-1)*Binomial(i,m)))))); # Muniru A Asiru, Jul 25 2018

A059110:= func< n,k | n eq 0 select 1 else Factorial(n-1)*Binomial(n,k)*Evaluate(LaguerrePolynomial(n-1, 1-k), -1) >;
[A059110(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 23 2021

Lprime := proc(n,i)
    if n = 0 and i = 0 then
        1;
    elif k = 0 then
        0 ;
    else
        n!/i!*binomial(n-1,i-1) ;
    end if;
end proc:
A059110 := proc(n,k)
    add(Lprime(n,i)*binomial(i,k),i=0..n) ;
end proc: # R. J. Mathar, Mar 15 2013

(* First program *)
lp[n_, i_] := Binomial[n-1, i-1]*n!/i!; lp[0, 0] = 1; t[n_, m_] := Sum[lp[n, i]*Binomial[i, m], {i, 0, n}]; Table[t[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Mar 26 2013 *)
(* Second program *)
A059110[n_, k_]:= If[n==0, 1, (n-1)!*Binomial[n, k]*LaguerreL[n-1, 1-k, -1]];
Table[A059110[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 23 2021 *)

def A059110(n, k): return 1 if n==0 else factorial(n-1)*binomial(n, k)*gen_laguerre(n-1, 1-k, -1)
flatten([[A059110(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 23 2021

E.g.f. for column m: (1/m!)*(x/(1-x))^m*e^(x/(x-1)), m >= 0.
From Wolfdieter Lang, Jun 22 2017: (Start)
E.g.f. for row polynomials in powers of x (e.g.f. of the triangle): exp(z/(1-z))* exp(x*z/(1-z)) (exponential Riordan).
Recurrence: T(n, 0) = Sum_{j=0} z(j)*T(n-1, j), n >= 1, with z(n) = (-1)^(n+1)*A028310(n), T(0, 0) = 1, T(n, m) = 0 n < m, T(n, m) = n*(T(n-1, m-1)/m + T(n-1, m)), n >= m >= 1 (from the z- and a-sequence, see a comment above).
Meixner type recurrence for the (monic) row polynomials R(n, x) = Sum_{m=0..n} T(n, m)*x^m: Sum_{k=0..n-1} (-1)^k*D^(k+1)*R(n, x) = n*R(n-1, x), n >=1, R(0, x) = 1, with D = d/dx.
General Sheffer recurrence: R(n, x) = (x+1)*(1+D)^2*R(n-1, x), n >=1, R(0, x) = 1.
(End)
P_n(x) = L_n(1+x) = n!*Lag_n(-(1+x);1), where P_n(x) are the row polynomials of this entry; L_n(x), the Lah polynomials of A105278; and Lag_n(x;1), the Laguerre polynomials of order 1. These relations follow from the relation between the iterated operator (x^2 D)^n and ((1+x)^2 D)^n with D = d/dx. - Tom Copeland, Jul 18 2018
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = (n-1)!*binomial(n, k)*LaguerreL(n-1, 1-k, -1) with T(0, 0) = 1.
Sum_{k=0..n} T(n, k) = A052897(n). (End)

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)

A059115 Expansion of e.g.f.: ((1-x)/(1-2x))exp(x/(1-x)).

Original entry on oeis.org

1, 2, 9, 58, 485, 4986, 60877, 861554, 13878153, 250854130, 5030058161, 110837000682, 2662669300909, 69270266115818, 1940260799150325, 58220372514830626, 1863293173842259217, 63356877145370671074

Offset: 0

Vladeta Jovovic, Jan 06 2001

L'(n,i) are unsigned Lah numbers (Cf. A008297): L'(n,i) = (n!/i!)*binomial(n-1,i-1) for i >= 1, L'(0,0) = 1, L'(n,0) = 0 for n > 0.

			(1-x)/(1-2*x)*exp(x/(1-x)) = 1 + 2*x + 9/2*x^2 + 29/3*x^3 + 485/24*x^4 + 831/20*x^5 + ...

G. C. Greubel, Table of n, a(n) for n = 0..250
Index entries for sequences related to Laguerre polynomials

Cf. A001861, A049020, A052897, A059099, A059110.

Row sums of A059114.

[Factorial(n)*(&+[Evaluate(LaguerrePolynomial(n-k, k-1), -1) : k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 23 2021

s := series((1-x)/(1-2*x)*exp(x/(1-x)), x, 21): for i from 0 to 20 do printf(`%d,`,i!*coeff(s,x,i)) od:

With[{nn=20},CoefficientList[Series[(1-x)/(1-2x) Exp[x/(1-x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 18 2020 *)
Table[n!*Sum[LaguerreL[n-k, k-1, -1], {k,0,n}], {n,0,30}] (* G. C. Greubel, Feb 23 2021 *)

{a(n)=if(n<0, 0, n!*polcoeff( (1-x)/(1-2*x)*exp(x/(1-x)+x*O(x^n)), n))} /* Michael Somos, Aug 03 2006 */

{a(n)=local(A); if(n<0,0, n++; A=vector(n); A[n]=1; for(k=1,n-1, A[n-k]=1; if(k>1, A[n-k+1]=A[n-k+2]); for(i=n-k+1,n, A[i]=A[i-1]+k*A[i])); A[n])} /* Michael Somos, Aug 03 2006 */

a(n) = n!*sum(k=0, n, pollaguerre(n-k, k-1, -1)); \\ Michel Marcus, Feb 23 2021

[factorial(n)*sum( gen_laguerre(n-k, k-1, -1) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Feb 23 2021

Sum_{m=0..n} Sum_{i=0..n} L'(n, i)*Product_{j=1..m} (i-j+1).
Given g.f. A(x), then g.f. A000522 = A(x/(1+x)). - Michael Somos, Aug 03 2006
a(n) = n!*Sum_{k=0..n} LaguerreL(n-k, k-1, -1). - G. C. Greubel, Feb 23 2021
a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(n + 1/2) / exp(n-1). - Vaclav Kotesovec, Feb 23 2021

Definition clarified by Harvey P. Dale, Jul 18 2020

A255806 Expansion of e.g.f.: exp(Sum_{k>=1} 3*x^k).

Original entry on oeis.org

1, 3, 15, 99, 801, 7623, 83079, 1017495, 13808097, 205374123, 3318673599, 57845821707, 1081091446785, 21553820597871, 456410531639799, 10225931132021247, 241609515712343361, 6002109578246918355, 156360266121378584943, 4261404847790207796147

Offset: 0

Vaclav Kotesovec, Mar 07 2015

In general, if e.g.f. = exp(Sum_{k>=1} m*x^k) = exp(m*x/(1-x)) and m>0, then a(n) ~ n! * m^(1/4) * exp(2*sqrt(m*n) - m/2) / (2 * sqrt(Pi) * n^(3/4)).

G. C. Greubel, Table of n, a(n) for n = 0..435
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A052897, A253286.

[n eq 0 select 1 else 3*Factorial(n-1)*Evaluate(LaguerrePolynomial(n-1, 1), -3): n in [0..25]]; // G. C. Greubel, Feb 24 2021

nmax=20; CoefficientList[Series[Exp[Sum[3*x^k,{k,1,nmax}]],{x,0,nmax}],x] * Range[0,nmax]!
CoefficientList[Series[E^(3*x/(1-x)), {x, 0, 20}], x] * Range[0, 20]!
Table[If[n==0, 1, 3*(n-1)!*LaguerreL[n-1, 1, -3]], {n, 0, 25}] (* G. C. Greubel, Feb 24 2021 *)

my(x='x +O('x^50)); Vec(serlaplace(exp(3*x/(1-x)))) \\ G. C. Greubel, Feb 05 2017

[1 if n==0 else 3*factorial(n-1)*gen_laguerre(n-1, 1, -3) for n in (0..25)] # G. C. Greubel, Feb 24 2021

E.g.f.: exp(3*x/(1-x)).
a(n) ~ 3^(1/4) * exp(2*sqrt(3*n) - 3/2) * n! / (2*sqrt(Pi)*n^(3/4)).
a(n) = (2*n+1)*a(n-1) - (n-2)*(n-1)*a(n-2). - Vaclav Kotesovec, Nov 04 2016
From G. C. Greubel, Feb 24 2021: (Start)
a(n) = A253286(n+3, 3).
a(n) = 3*(n-1)!*LaguerreL(n-1, 1, -3) with a(0) = 1. (End)
For n > 0, a(n) = (n-1)! * Sum_{k=1..n} binomial(n,k) * 3^k / (k-1)!. - Vaclav Kotesovec, Aug 24 2025

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

A059114 Triangle T(n,m)= Sum_{i=0..n} L'(n,i)*Product_{j=1..m} (i-j+1), read by rows.

Original entry on oeis.org

1, 1, 1, 3, 4, 2, 13, 21, 18, 6, 73, 136, 156, 96, 24, 501, 1045, 1460, 1260, 600, 120, 4051, 9276, 15030, 16320, 11160, 4320, 720, 37633, 93289, 170142, 219450, 192360, 108360, 35280, 5040, 394353, 1047376, 2107448, 3116736, 3294480, 2405760, 1149120, 322560, 40320

Offset: 0

Vladeta Jovovic, Jan 04 2001

L'(n,i) are unsigned Lah numbers (Cf. A008297): L'(n,i) = (n!/i!)*binomial(n-1,i-1) for i >= 1, L'(0,0) = 1, L'(n,0) = 0 for n > 0.

			Triangle begins as:
    1;
    1,    1;
    3,    4,    2;
   13,   21,   18,    6;
   73,  136,  156,   96,  24;
  501, 1045, 1460, 1260, 600, 120;
  ...;
E.g.f. for T(n, 2) = (x/(1-x))^2*e^(x/(x-1)) = x^2 + 3*x^3 + 13/2*x^4 + 73/6*x^5 + 167/8*x^6 + 4051/120*x^7 + ...

Cf. A000262, A052852, A052897, A059110.

Row sums give A059115. Alternating row sums give A288268.

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

Table[n!*LaguerreL[n-k, k-1, -1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 23 2021 *)

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

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

E.g.f. for T(n, k) = (x/(1-x))^k * exp(x/(x-1)).
T(n, k)= Sum_{i=0..n} L'(n,i) * ( Product_{j=1..k} (i-j+1) ).
T(n, 0) = A000262(n).
T(n, 1) = A052852(n).
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = n! * k! * Sum_{j=0..n} binomial(j, k)*binomial(n-1, j-1)/j!.
T(n, k) = n! * Laguerre(n-k, k-1, -1).
T(n, k) = n!*binomial(n-1, k-1)*Hypergeometric1F1([k-n], [k], -1) with T(n, 0) = Hypergeometric2F0([1-n, -n], [], 1). (End)

A086915 Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).

Original entry on oeis.org

2, 4, 4, 12, 24, 8, 48, 144, 96, 16, 240, 960, 960, 320, 32, 1440, 7200, 9600, 4800, 960, 64, 10080, 60480, 100800, 67200, 20160, 2688, 128, 80640, 564480, 1128960, 940800, 376320, 75264, 7168, 256, 725760, 5806080, 13547520, 13547520, 6773760, 1806336

Offset: 1

Vladeta Jovovic, Sep 24 2003

Also the Bell transform of A052849(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

The coefficients of n! * L_n(-2*x,-1), where n! * L_n(-x,-1) are the normalized, unsigned Laguerre polynomials of order -1 of A105278, also known as the Lah polynomials, which are also a shifted version of n! * L_n(-x,1). Cf. p. 8 of the Gross and Matytsin link. - Tom Copeland, Sep 30 2016

			Triangle begins:
   2;
   4,   4;
  12,  24,  8;
  48, 144, 96, 16;
  ...

G. C. Greubel, Rows n=1..100 of the triangle, flattened
D. Gross and A. Matytsin, Instanton induced large N phase transitions in two and four dimensional QCD, arXiv:hep-th/9404004, 1994.
Index entries for sequences related to Laguerre polynomials

Cf. A008297, A052897 (row sums), A059110, A079621, A105278.

[Factorial(n)*Binomial(n-1,k-1)*2^k/Factorial(k): k in [1..n], n in [1..10]]; // G. C. Greubel, May 23 2018

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ...) as column 0.
BellMatrix(n -> 2*(n+1)!, 9); # Peter Luschny, Jan 26 2016

Flatten[Table[n!/k! Binomial[n-1,k-1]2^k,{n,10},{k,n}]] (* Harvey P. Dale, May 25 2011 *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[2*(#+1)!&, rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

for(n=1,10, for(k=1, n, print1(n!/k!*binomial(n-1,k-1)*2^k, ", "))) \\ G. C. Greubel, May 23 2018

E.g.f.: exp(2*x*y/(1-x)).
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = (-2)^k * A008297(n, k) = 2^k * A105278(n, k).
Sum_{k=1..n} T(n, k) = 2 * n! * Hypergeometric1F1([1-n], [2], -2) = 2*(n-1)! * LaguerreL(n-1, 1, -2) = A253286(n, 2). (End)

A052838 Expansion of e.g.f.: (exp(x/(1-x)) - 1)^2.

Original entry on oeis.org

0, 0, 2, 18, 158, 1510, 15962, 186270, 2385182, 33290862, 503277242, 8193803926, 142938943886, 2659770747270, 52581058479770, 1100423513438766, 24302677755662654, 564770268904566238

Offset: 0

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

Previous name was: A simple grammar.

G. C. Greubel, Table of n, a(n) for n = 0..250
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 805
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A052897.

[0] cat [2*Factorial(n)*(&+[Binomial(n-1,j)*(2^j-1)/Factorial(j+1): j in [0..n-1]]) : n in [1..25]]; // G. C. Greubel, Feb 23 2021

spec := [S,{C=Sequence(Z,1 <= card),B=Set(C,1 <= card),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[(E^(x/(1-x))-1)^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)
Table[n!*(LaguerreL[n, -1, -2] - 2* LaguerreL[n, -1, -1]) + Boole[n==0], {n,0,20}] (* G. C. Greubel, Feb 23 2021 *)

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

E.g.f.: (exp(-x/(-1+x)) - 1)^2.
Recurrence: n*(2 +5*n +4*n^2 +n^3)*a(n) - (18 +35*n +21*n^2 +4*n^3)*a(n+1) +2*(19 +15*n +3*n^2)*a(n+2) - (13 +4*n)*a(n+3) + a(n+4) = 0, with a(1)=0, a(0)=0, a(2)=2, a(3)=18.
From Vaclav Kotesovec, Sep 30 2013: (Start)
a(n) = A052897(n) - 2*A000262(n) for n > 0.
a(n) ~ 2^(-1/4)*exp(2*sqrt(2*n)-n-1)*n^(n-1/4). (End)
From G. C. Greubel, Feb 23 2021: (Start)
a(n) = 2 * n! * Sum_{j=0..n-1} binomial(n-1, j)*(2^j -1)/(j+1)!.
a(n) = n! * (LaguerreL(n, -1, -2) - 2*LaguerreL(n, -1, -1)) + [n=0]. (End)

New name, using e.g.f., by Vaclav Kotesovec, Sep 30 2013

A317364 Expansion of e.g.f. exp(2*x/(1 + x)).

Original entry on oeis.org

1, 2, 0, -4, 16, -48, 64, 800, -12288, 127232, -1150976, 9266688, -58726400, 68777984, 7510646784, -207794409472, 4241007640576, -77359570944000, 1321952191971328, -21274345818161152, 313768799799607296, -3838962981483839488, 21775623343518515200, 859024717017756205056

Offset: 0

Ilya Gutkovskiy, Jul 26 2018

sign

Inverse Lah transform of the powers of 2 (A000079).

Robert Israel, Table of n, a(n) for n = 0..451
N. J. A. Sloane, Transforms

Cf. A000079, A052897, A111884.

Cf. A001263, A008297, A086915.

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

a:= proc(n) option remember; add((-1)^(n-k)*
      n!/k!*binomial(n-1, k-1)*2^k, k=0..n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2018

nmax = 23; CoefficientList[Series[Exp[2 x/(1 + x)], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[Exp[-2 (-x)^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] 2^k n!/k!, {k, 0, n}], {n, 0, 23}]
Join[{1}, Table[2 (-1)^(n+1) n! Hypergeometric1F1[1-n, 2, 2], {n, 23}]]

a(n) = if (n==0, 1, (-1)^n*n!*pollaguerre(n, -1, 2)); \\ Michel Marcus, Feb 23 2021

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

E.g.f.: Product_{k>=1} exp(-2*(-x)^k).
a(n) = 2*(-1)^(n+1) * n! * Hypergeometric1F1([1-n], [2], 2).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*2^k*n!/k!.
(n^2 + n)*a(n) + 2*n*a(n+1) + a(n+2) = 0. - Robert Israel, Aug 18 2019
From G. C. Greubel, Feb 23 2021: (Start)
a(n) = (-1)^n * n! * Laguerre(n, -1, 2) for n > 0 with a(0) = 1.
a(n) = Sum_{k=0..n} (-1)^(n-k) * A086915(n, k).
a(n) = (-1)^n * Sum_{k=0..n} 2^k * A008297(n, k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * (n-k+1)! * A001263(n, k). (End)

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

A059110 Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A059115 Expansion of e.g.f.: ((1-x)/(1-2*x))*exp(x/(1-x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

PARI

Sage

Formula

Extensions

A255806 Expansion of e.g.f.: exp(Sum_{k>=1} 3*x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

A059110 Triangle T = A007318A271703; T(n,m)= Sum_{i=0..n} L'(n,i)binomial(i,m), m=0..n.

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.

A059115 Expansion of e.g.f.: ((1-x)/(1-2x))exp(x/(1-x)).