A025168 -id:A025168 - cp's OEIS frontend

A103446 Unlabeled analog of A025168.

0, 1, 3, 8, 21, 54, 137, 344, 856, 2113, 5179, 12614, 30548, 73595, 176455, 421215, 1001388, 2371678, 5597245, 13166069, 30873728, 72185937, 168313391, 391428622, 908058205, 2101629502, 4853215947, 11183551059, 25718677187, 59030344851, 135237134812

Offset: 0

Thomas Wieder, Feb 06 2005; revised Feb 20 2006

nonn

Or, if the initial 0 is omitted, this is the binomial transform of the partition numbers p(1), p(2), ... = 1, 2, 3, 5, 7, 11, 15, 22, 30, ... (A000041 without the initial 1).
The most precise definition of this sequence is the Maple combstruct command given below. See the first Wieder link for further details.
Sequence appears to have a rational o.g.f. - Ralf Stephan, May 18 2007
For n>0, row sums of triangle A137151. - Gary W. Adamson, Jan 23 2008
a(n) = A218482(n) for n>=1; see A218482 for more formulas.

			Let {} denote a set, [] a list and Z an unlabeled element.
a(3) = 8 because we have {[[Z]],[[Z]],[[Z]]}, {[[Z],[Z]],[[Z]]}, {[[Z],[Z],[Z]]}, {[[Z],[Z,Z]]}, {[[Z,Z],[Z]]}, {[[Z,Z]],[[Z]]}, {[[Z]],[[Z,Z]]}, {[[Z,Z,Z]]}.

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.
Thomas Wieder, Expanded definitions of A103446 and A025168

Cf. A025168, A034691, A050351, A137151, A185003 (log), A218482.

with(combstruct); SubSetSeqU := [T,{T=Subst(U,S),S=Set(U,card>=1),U=Sequence(Z,card>=1)},unlabeled]; [seq(count(SubSetSeqU, size=n), n=0..30)];
allstructs(SubSetSeq,size=3); # to get the structures for n=3 - this output is shown in the example lines.

Flatten[{0, Table[Sum[Binomial[n-1,k]*PartitionsP[k+1],{k,0,n-1}],{n,1,30}]}] (* Vaclav Kotesovec, Jun 25 2015 *)

{a(n)=if(n<1,0,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x+x*O(x^n))^m/m)),n))} \\ Paul D. Hanna, Apr 21 2010

{a(n)=if(n<1,0,polcoeff(exp(sum(m=1,n,x^m/m*sum(k=1,m,binomial(m,k)*sigma(k)))+x*O(x^n)),n))} \\ Paul D. Hanna, Feb 04 2012

Vec(1/eta('x/(1-'x)+O('x^66))) \\ Joerg Arndt, Jul 30 2011

O.g.f.: exp( Sum_{n>=1} sigma(n)*x^n/(1-x)^n/n ) - 1. - Paul D. Hanna, Apr 21 2010
O.g.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=1..n} binomial(n,k)*sigma(k) ) - 1. - Paul D. Hanna, Feb 04 2012
O.g.f. P(x/(1-x)), where P(x) is the o.g.f. for number of partitions (A000041) a(n)=sum_{k=1,n} ( binomial(n-1,k-1)*A000041(k)). - Vladimir Kruchinin, Aug 10 2010
a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-2) / (n*sqrt(3)). - Vaclav Kotesovec, Jun 25 2015

I can confirm that the terms shown are the binomial transform of the partition sequence 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, ... (A000041 without the a(0) term). - N. J. A. Sloane, May 18 2007

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

A025167 E.g.f: exp(x/(1-2x))/(1-2x).

Original entry on oeis.org

1, 3, 17, 139, 1473, 19091, 291793, 5129307, 101817089, 2250495523, 54780588561, 1455367098923, 41888448785857, 1298019439099059, 43074477771208913, 1523746948247663611, 57229027745514785793, 2274027983943883110467

Offset: 0

Wouter Meeussen

nonn

Polynomials in A021009 evaluated at -2.

Also, a(n) is the number of signed permutations of length 2n that are equal to their reverse-complements and avoid the pattern (-2,-1). As a result, a(n) also gives the same thing but for avoiding any one of (-1,-2), (+2,+1) or (+1,+2) instead of (-2,-1) (see the Hardt and Troyka reference). - Justin M. Troyka, Aug 05 2011

			Since a(2) = 17, there are 17 signed permutations of 4 that are equal to their reverse-complements and avoid (-2,-1).  Some of these are: (+1,+3,+2,+4), (+2,-1,-4,+3), (+3,-1,-4,+2), (-1,-2,-3,-4). - _Justin M. Troyka_, Aug 05 2011

Seiichi Manyama, Table of n, a(n) for n = 0..399
A. Hardt and J. M. Troyka, Restricted symmetric signed permutations, Pure Mathematics and Applications, Vol. 23 (No. 3, 2012), pp. 179--217.
A. Hardt and J. M. Troyka, Slides (associated with the Hardt and Troyka reference above).

Cf. A025166, A025168, A002720.

a := n -> (-2)^n*KummerU(-n, 1, -1/2):
seq(simplify(a(n)), n=0..17); # Peter Luschny, Feb 12 2020

Table[ n! 2^n LaguerreL[ n, -1/2 ], {n, 0, 12} ]
f[n_] := Sum[k!*2^k*Binomial[n, k]^2, {k, 0, n}]; Table[ f[n], {n, 0, 17}] (* Robert G. Wilson v, Mar 16 2005 *)
a = {1, 3}; For[n = 2, n < 13, n++, a = Append[a, (4 n - 1) a[[n]] - 4 (n - 1)^2 a[[n - 1]]]]; a  (* Justin M. Troyka, Aug 05 2011 *)

{a(n)=n!^2*polcoeff(exp(2*x+x*O(x^n))*sum(m=0,n,x^m/m!^2),n)}

a(n) = Sum_{k=0..n} k!*2^k*binomial(n, k)^2. - Robert G. Wilson v, Mar 16 2005 [corrected by Ilya Gutkovskiy, Oct 01 2018]
a(n) = Sum_{k=0..n-1} 2^{n-1-k}*[(n-1)! ]^2/[(k!)^2*(n-1-k)! ]. - Huajun Huang (huanghu(AT)auburn.edu), Oct 10 2005
a(0) = 1; a(1) = 3; a(n) = (4n-1) * a(n-1) - 4 (n-1)^2 * a(n-2) for n >= 2. - Justin M. Troyka, Aug 05 2011
E.g.f.: exp(2*x) * Sum_{n>=0} x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. - Paul D. Hanna, Nov 18 2011
a(n) ~ n^(n+1/4)*2^(n-1/4)*exp(-n+sqrt(2*n)-1/4) * (1 + sqrt(2)/(3*sqrt(n))). - Vaclav Kotesovec, Jun 22 2013
a(n) = (-2)^n*KummerU(-n, 1, -1/2). - Peter Luschny, Feb 12 2020

More terms from Vladeta Jovovic, Jan 29 2003

A321837 Expansion of e.g.f.: exp(x/(1-3*x)).

Original entry on oeis.org

1, 1, 7, 73, 1009, 17341, 355951, 8488117, 230439553, 7013527129, 236419161751, 8740611892321, 351566026652017, 15280473017519893, 713558666964639679, 35623071889296787981, 1893073661362838712961, 106682309871314293118257

Offset: 0

Ludovic Schwob, Nov 19 2018

nonn

For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.

Ludovic Schwob, Table of n, a(n) for n = 0..199
Norihiro Nakashima, Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A000262, A025168, A321847, A321848, A321849, A321850 (analogs for k=1,2,4,5,6,7).

Concatenation([1], List([1..25], n-> Sum([1..n], k-> 3^(n-k)*(Factorial(n)/Factorial(k))*Binomial(n-1, k-1)))); # G. C. Greubel, Dec 14 2018

[1] cat [&+[3^(n-k)*Factorial(n) div Factorial(k)*Binomial(n-1, k-1):  k in [0..n]]: n in [1.. 18]]; // Vincenzo Librandi, Dec 08 2018

seq(coeff(series(factorial(n)*exp(x/(1-3*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018

a[n_] := Sum[3^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (6n - 5)*a[n - 1] - 9(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)

my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-3*x)))) \\ Michel Marcus, Nov 25 2018

{c[1]:c[0]*factorial(c[1]) for c in (exp(x/(1-3*x))).taylor(x,0,25).coefficients()} # G. C. Greubel, Dec 14 2018

a(n) = Sum_{k=0..n} 3^(n-k)*(n!/k!)*binomial(n-1, k-1).
Recurrence: a(n) = (6*n-5)*a(n-1) - 9*(n-2)*(n-1)*a(n-2).
a(n) ~ n! * exp(2*sqrt(n/3) - 1/6) * 3^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018

A321847 E.g.f.: exp(x/(1 - 4*x)).

Original entry on oeis.org

1, 1, 9, 121, 2161, 48081, 1279801, 39631369, 1398961761, 55422807841, 2434261023721, 117366299630361, 6161301265353169, 349768597919934961, 21347094823271661081, 1393695557886847095721, 96910923898115717350081, 7149718240571434690591809

Offset: 0

Ludovic Schwob, Nov 19 2018

nonn

For k = 2, 3, 4, ... the difference a(n+k) - a(n) is divisible by k.

Ludovic Schwob, Table of n, a(n) for n = 0..199
Norihiro Nakashima, Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A000262, A025168, A321837, A321848, A321849, A321850.

seq(coeff(series(factorial(n)*exp(x/(1-4*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018

a[n_] := Sum[4^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (8n - 7)*a[n - 1] - 16(n - 2)(n - 1) *a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)
CoefficientList[Series[Exp[x/(1 - 4*x)], {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Dec 07 2018 *)

(a[0] : 1, a[1] : 1, a[n] := (8*n - 7)*a[n-1] - 16*(n-2)*(n-1)*a[n-2], makelist(a[n], n, 0, 20)); /* Franck Maminirina Ramaharo, Nov 27 2018 */

my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-4*x)))) \\ Michel Marcus, Nov 25 2018

a(n) = Sum_{k=0..n} 4^(n - k)*(n!/k!)*binomial(n-1, k-1).
Recurrence: a(n) = (8*n - 7)*a(n-1) - 16*(n-2)*(n-1)*a(n-2).
a(n) ~ n! * exp(sqrt(n) - 1/8) * 2^(2*n - 3/2) / (sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018

A321848 E.g.f.: exp(x/(1-5*x)).

Original entry on oeis.org

1, 1, 11, 181, 3961, 108101, 3532651, 134415961, 5834249681, 284391878761, 15378011541451, 913297438474301, 59086483931657161, 4135583008765323181, 311324086814794408811, 25079793551003791168801, 2152597370423901820231201, 196089415332225446044417361

Offset: 0

Ludovic Schwob, Nov 19 2018

nonn

For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.

Ludovic Schwob, Table of n, a(n) for n = 0..199
Norihiro Nakashima, Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A000262, A025168, A321837, A321847, A321849, A321850.

seq(coeff(series(factorial(n)*exp(x/(1-5*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018

a[n_] := Sum[5^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (10 n - 9)*a[n - 1] - 25(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)
With[{nn=20},CoefficientList[Series[Exp[x/(1-5x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 19 2024 *)

my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-5*x)))) \\ Michel Marcus, Nov 25 2018

a(n) = Sum_{k=0..n} 5^(n-k)*(n!/k!)*binomial(n-1, k-1).
Recurrence: a(n) = (10*n-9)*a(n-1) - 25*(n-2)*(n-1)*a(n-2).
a(n) ~ n! * exp(2*sqrt(n/5) - 1/10) * 5^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018

A321849 Expansion of e.g.f.: exp(x/(1-6*x)).

Original entry on oeis.org

1, 1, 13, 253, 6553, 211801, 8201701, 369979093, 19047250993, 1101705494833, 70715424362941, 4987040544656941, 383243311962126793, 31871863566298601353, 2851588139929576342933, 273093945561709965890821, 27871997808801906673665121

Offset: 0

Ludovic Schwob, Nov 19 2018

nonn

For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.

Ludovic Schwob, Table of n, a(n) for n = 0..199

Cf. A000262, A025168, A321837, A321847, A321848, A321850.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1-6*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Dec 08 2018

[1] cat [&+[6^(n-k)* Factorial(n)/Factorial(k)*Binomial(n-1, k-1):  k in [0..n]]: n in [1.. 18]]; // Vincenzo Librandi, Dec 08 2018

seq(coeff(series(factorial(n)*exp(x/(1-6*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018

a[n_] := Sum[6^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (12n - 11)*a[n - 1] - 36(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)

my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-6*x)))) \\ Michel Marcus, Nov 25 2018

f= exp(x/(1-6*x))
g=f.taylor(x,0,13)
L=g.coefficients()
coeffs={c[1]:c[0]*factorial(c[1]) for c in L}
coeffs  # G. C. Greubel, Dec 08 2018

a(n) = Sum_{k=0..n} 6^(n-k)*(n!/k!)*binomial(n-1, k-1).
Recurrence: a(n) = (12*n-11)*a(n-1) - 36*(n-2)*(n-1)*a(n-2).
a(n) ~ n! * exp(sqrt(2*n/3) - 1/12) * 6^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018

A321850 E.g.f.: exp(x/(1-7*x)).

Original entry on oeis.org

1, 1, 15, 337, 10081, 376461, 16849351, 878797165, 52324954977, 3501300491641, 260062721279551, 21228108532279881, 1888618754806601665, 181873163575529411077, 18846187172580219099831, 2090754000231731874682021, 247221828043044971020645441

Offset: 0

Ludovic Schwob, Nov 19 2018

nonn

For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.

Ludovic Schwob, Table of n, a(n) for n = 0..199

Cf. A000262, A025168, A321837, A321847, A321848, A321849.

[1] cat [&+[7^(n-k)*Factorial(n)/Factorial(k)*Binomial(n-1, k-1):  k in [0..n]]: n in [1.. 18]]; // Vincenzo Librandi, Dec 08 2018

seq(coeff(series(factorial(n)*exp(x/(1-7*x)),x,n+1), x, n), n = 0 .. 17); # Muniru A Asiru, Nov 24 2018

a[n_] := Sum[7^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (14n - 13)*a[n - 1] - 49(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)

my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-7*x)))) \\ Michel Marcus, Nov 25 2018

a(n) = Sum_{k=0..n} 7^(n-k)*(n!/k!)*binomial(n-1, k-1).
Recurrence: a(n) = (14*n-13)*a(n-1) - 49*(n-2)*(n-1)*a(n-2).
a(n) ~ n! * exp(2*sqrt(n/7) - 1/14) * 7^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Nov 21 2018

A293146 a(n) = n! * [x^n] exp(x/(1 - n*x)).

Original entry on oeis.org

1, 1, 5, 73, 2161, 108101, 8201701, 878797165, 126422091713, 23514740267401, 5492576235204901, 1574136880033408241, 543143967119720304625, 222106209904092987888013, 106221716052645457812866501, 58741017143127754662557082901, 37194600833984874761008613195521

Offset: 0

Ilya Gutkovskiy, Oct 01 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..232

Cf. A000262, A025168, A293145.

S:=series(exp(x/(1-n*x)),x,31):
seq(coeff(S,x,n)*n!,n=0..30); # Robert Israel, Oct 01 2017

Table[n! SeriesCoefficient[Exp[x/(1 - n x)], {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[n! SeriesCoefficient[Product[Exp[n^k x^(k + 1)], {k, 0, n}], {x, 0, n}], {n, 1, 16}]]
Join[{1}, Table[Sum[n^(n - k) n!/k! Binomial[n - 1, k - 1], {k, n}], {n, 1, 16}]]
Join[{1}, Table[n^n (n - 1)! Hypergeometric1F1[1 - n, 2, -1/n], {n, 1, 16}]]

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

a(n) ~ BesselI(1, 2) * sqrt(2*Pi) * n^(2*n-1/2) / exp(n). - Vaclav Kotesovec, Oct 01 2017

a(n) = n! * Sum_{k=1..n} n^(n-k) * binomial(n-1,k-1)/k! for n > 0. - Seiichi Manyama, Feb 03 2021

A025166 E.g.f.: -exp(-x/(1-2x))/(1-2x).

Original entry on oeis.org

-1, -1, -1, 7, 127, 1711, 23231, 334391, 5144063, 84149983, 1446872959, 25661798119, 454494403199, 7489030040207, 89680375568447, -759618144120809, -127049044802971649, -7480338932613448769, -369274690558092738817, -17262533154073740329017

Offset: 0

Wouter Meeussen

sign

Polynomials in A021009 evaluated at 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A025167, A025168.

a := n -> -(-2)^n*KummerU(-n, 1, 1/2):
seq(simplify(a(n)), n=0..19); # Peter Luschny, Feb 12 2020

Table[ -n! 2^n LaguerreL[ n, 1/2 ], {n, 0, 12} ]

Conjecture: a(n) + (-4*n+3)*a(n-1) + 4*(n-1)^2*a(n-2) = 0. - R. J. Mathar, Feb 05 2013
a(n) = -(-2)^n*KummerU(-n, 1, 1/2). - Peter Luschny, Feb 12 2020
Sum_{n>=0} a(n) * x^n / (n!)^2 = -exp(2*x) * BesselJ(0,2*sqrt(x)). - Ilya Gutkovskiy, Jul 17 2020

Corrected and extended by Vladeta Jovovic, Jan 29 2003

cp's OEIS Frontend

A103446 Unlabeled analog of A025168.

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

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A025167 E.g.f: exp(x/(1-2*x))/(1-2*x).

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A321837 Expansion of e.g.f.: exp(x/(1-3*x)).

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A321847 E.g.f.: exp(x/(1 - 4*x)).

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A321848 E.g.f.: exp(x/(1-5*x)).

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A025167 E.g.f: exp(x/(1-2x))/(1-2x).

A025166 E.g.f.: -exp(-x/(1-2x))/(1-2x).