A187735 -id:A187735 - cp's OEIS frontend

A014479 Exponential generating function = (1+2x)/(1-2x)^3.

1, 8, 72, 768, 9600, 138240, 2257920, 41287680, 836075520, 18579456000, 449622835200, 11771943321600, 331576403558400, 9998303861145600, 321374052679680000, 10969567664799744000, 396275631890890752000

Offset: 0

N. J. A. Sloane

nonn

Robert Israel, Table of n, a(n) for n = 0..395
Ben Adenbaum, Jennifer Elder, Pamela E. Harris, and J. Carlos Martínez Mori, Boolean intervals in the weak Bruhat order of a finite Coxeter group, arXiv:2403.07989 [math.CO], 2024. See p. 8.

Cf. A001620, A014477, A187735.

seq(add(count(Composition(k))*count(Permutation(k)),k=1..n),n=1..17); # Zerinvary Lajos, Oct 17 2006
seq(2^n*(n+1)^2*n!, n=0..30); # Robert Israel, Oct 28 2015

Table[2^n (n+1)^2 n!, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)

{a(n)=polcoeff( sum(m=0,n,(2*m+1)^(m+1)*x^m / (1 + (2*m+1)*x +x*O(x^n))^(m+1)),n)} \\ Paul D. Hanna, Jan 02 2013
for(n=0,20,print1(a(n),", "))

vector(30, n, n--; n!*(n+1)^2*2^n) \\ Altug Alkan, Oct 28 2015

a(n) = A014477(n) * n!. - Franklin T. Adams-Watters, Nov 02 2006
G.f.: Sum_{n>=0} (2*n+1)^(n+1) * x^n / (1 + (2*n+1)*x)^(n+1). - Paul D. Hanna, Jan 02 2013
From Vladimir Reshetnikov, Oct 28 2015: (Start)
a(n) = 2^n*(n+1)^2*n!.
Recurrence: a(0) = 1, n*a(n) = 2*(n+1)^2*a(n-1). (End)
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = 2*(Ei(1/2) - gamma + log(2)), where Ei(x) is the exponential integral and gamma is Euler's constant (A001620).
Sum_{n>=0} (-1)^n/a(n) = 2*(gamma - Ei(-1/2) - gamma - log(2)). (End)

A187738 G.f.: Sum_{n>=0} (3n+1)^n x^n / (1 + (3n+1)x)^n.

Original entry on oeis.org

1, 4, 33, 378, 5508, 97200, 2012040, 47764080, 1278607680, 38093690880, 1249949232000, 44783895340800, 1739500776921600, 72804471541401600, 3266273336880153600, 156364149105964800000, 7955807906511489024000, 428712969452770050048000, 24390705726366524633088000

Offset: 0

Paul D. Hanna, Jan 03 2013

nonn

 More generally,
if Sum_{n>=0} a(n)*x^n = Sum_{n>=0} (b*n+c)^n * x^n / (1 + (b*n+c)*x)^n,
then Sum_{n>=0} a(n)*x^n/n! = (2 - 2*(b-c)*x + b*(b-2*c)*x^2)/(2*(1-b*x)^2)
so that a(n) = (b*n + (b+2*c)) * b^(n-1) * n!/2 for n>0 with a(0)=1.

			G.f.: A(x) = 1 + 4*x + 33*x^2 + 378*x^3 + 5508*x^4 + 97200*x^5 +...
where
A(x) = 1 + 4*x/(1+4*x) + 7^2*x^2/(1+7*x)^2 + 10^3*x^3/(1+10*x)^3 + 13^4*x^4/(1+13*x)^4 + 16^5*x^5/(1+16*x)^5 +...

Cf. A187735, A014479, A187739, A221160, A221161, A187740.

{a(n)=polcoeff(sum(m=0,n,((3*m+1)*x)^m/(1+(3*m+1)*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) = (3*n+5) * 3^(n-1) * n!/2 for n>0 with a(0)=1.

E.g.f.: (2 - 4*x + 3*x^2) / (2*(1-3*x)^2).

A187739 G.f.: Sum_{n>=0} (3n+2)^n x^n / (1 + (3n+2)x)^n.

Original entry on oeis.org

1, 5, 39, 432, 6156, 106920, 2187000, 51438240, 1366787520, 40474546560, 1321374902400, 47140942464000, 1824354473356800, 76113765702374400, 3405263691641011200, 162618715070203392000, 8256027072794941440000, 444024146933226123264000, 25217509310311152586752000

Offset: 0

Paul D. Hanna, Jan 03 2013

nonn

 More generally,
if Sum_{n>=0} a(n)*x^n = Sum_{n>=0} (b*n+c)^n * x^n / (1 + (b*n+c)*x)^n,
then Sum_{n>=0} a(n)*x^n/n! = (2 - 2*(b-c)*x + b*(b-2*c)*x^2)/(2*(1-b*x)^2)
so that a(n) = (b*n + (b+2*c)) * b^(n-1) * n!/2 for n>0 with a(0)=1.

			G.f.: A(x) = 1 + 5*x + 39*x^2 + 432*x^3 + 6156*x^4 + 106920*x^5 +...
where
A(x) = 1 + 5*x/(1+5*x) + 8^2*x^2/(1+8*x)^2 + 11^3*x^3/(1+11*x)^3 + 14^4*x^4/(1+14*x)^4 + 17^5*x^5/(1+17*x)^5 +...

Cf. A187735, A014479, A187738, A221160, A221161, A187740.

{a(n)=polcoeff(sum(m=0,n,((3*m+2)*x)^m/(1+(3*m+2)*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) = (3*n+7) * 3^(n-1) * n!/2 for n>0 with a(0)=1.

E.g.f.: (2 - 2*x - 3*x^2) / (2*(1-3*x)^2).

A187740 G.f.: Sum_{n>=0} (5n+1)^n x^n / (1 + (5n+1)x)^n.

Original entry on oeis.org

1, 6, 85, 1650, 40500, 1200000, 41625000, 1653750000, 74025000000, 3685500000000, 201993750000000, 12084187500000000, 783523125000000000, 54729675000000000000, 4097124281250000000000, 327237848437500000000000, 27775310062500000000000000, 2496585341250000000000000000

Offset: 0

Paul D. Hanna, Jan 03 2013

nonn

			G.f.: A(x) = 1 + 6*x + 85*x^2 + 1650*x^3 + 40500*x^4 + 1200000*x^5 +...
where
A(x) = 1 + 6*x/(1+6*x) + 11^2*x^2/(1+11*x)^2 + 16^3*x^3/(1+16*x)^3 + 21^4*x^4/(1+21*x)^4 + 26^5*x^5/(1+26*x)^5 +...

Cf. A187735, A014479, A187738, A187739, A221160, A221161.

{a(n)=polcoeff(sum(m=0, n, ((5*m+1)*x)^m/(1+(5*m+1)*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) = (5*n+7) * 5^(n-1) * n!/2 for n>0 with a(0)=1.

E.g.f.: (2 - 8*x + 15*x^2) / (2*(1-5*x)^2).

A187741 G.f.: Sum_{n>=0} (1 + nx)^n x^n / (1 + x + n*x^2)^n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 24, 60, 120, 360, 720, 2520, 5040, 20160, 40320, 181440, 362880, 1814400, 3628800, 19958400, 39916800, 239500800, 479001600, 3113510400, 6227020800, 43589145600, 87178291200, 653837184000, 1307674368000, 10461394944000, 20922789888000

Offset: 0

Paul D. Hanna, Jan 03 2013

nonn

This is an enumeration of the disjoint union (with repetition) of A001710(n), for n > 0, and A000142(n), for n > 0. The first lists the orders of the alternating groups; the second lists the orders of the symmetric groups. - Hal M. Switkay, Mar 13 2023

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 24*x^7 + 60*x^8 +...
where
A(x) = 1 + (1+x)*x/(1+x+x^2) + (1+2*x)^2*x^2/(1+x+2*x^2)^2 + (1+3*x)^3*x^3/(1+x+3*x^2)^3 + (1+4*x)^4*x^4/(1+x+4*x^2)^4 + (1+5*x)^5*x^5/(1+x+5*x^2)^5 +...

Cf. A000142, A001710, A187742, A187735, A208236, A204064, A361382.

a[n_] := If[OddQ[n], ((n + 1)/2)!, (n/2 + 1)!/2]; a[0] = 1; Array[a, 32, 0] (* Amiram Eldar, Dec 11 2022 *)

{a(n)=polcoeff( sum(m=0, n, (x+m*x^2)^m / (1 + x+m*x^2 +x*O(x^n))^m), n)}
for(n=0, 40, print1(a(n), ", "))

{a(n)=if(n==0,1, if(n%2==0, ((n+2)/2)!/2, ((n+1)/2)! ))}
for(n=0, 30, print1(a(n), ", "))

G.f.: 1/2 + (1+2*x) * Sum_{n>=0} (n+1)!*x^(2*n)/2.
a(2*n) = (n+1)!/2,  a(2*n-1) = n!,  for n>0 with a(0)=1.
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 3*e - 4.
Sum_{n>=0} (-1)^n/a(n) = e - 2. (End)

A221160 G.f.: Sum_{n>=0} (4n+1)^n x^n / (1 + (4n+1)x)^n.

Original entry on oeis.org

1, 5, 56, 864, 16896, 399360, 11059200, 350945280, 12551454720, 499415777280, 21879167385600, 1046394961920000, 54245114825932800, 3029690116944691200, 181363518724689100800, 11583863454028529664000, 786298610212845649920000, 56523637237014847291392000

Offset: 0

Paul D. Hanna, Jan 03 2013

nonn

			G.f.: A(x) = 1 + 5*x + 39*x^2 + 432*x^3 + 6156*x^4 + 106920*x^5 +...
where
A(x) = 1 + 5*x/(1+5*x) + 9^2*x^2/(1+9*x)^2 + 13^3*x^3/(1+13*x)^3 + 17^4*x^4/(1+17*x)^4 + 21^5*x^5/(1+21*x)^5 +...

Cf. A187735, A014479, A187738, A187739, A221161, A187740, A333419.

a[n_] := (2*n + 3)*4^(n - 1)*n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Dec 23 2022 *)

{a(n)=polcoeff(sum(m=0,n,((4*m+1)*x)^m/(1+(4*m+1)*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) = (2*n+3) * 4^(n-1) * n!  for n>0 with a(0)=1.
E.g.f.: (1 - 3*x + 4*x^2) / (1-4*x)^2.
From Amiram Eldar, Dec 23 2022: (Start)
Sum_{n>=0} 1/a(n) = 8*exp(1/4) - 1/3 - 8*sqrt(Pi)*erfi(1/2), where erfi is the imaginary error function.
Sum_{n>=0} (-1)^n/a(n) = 8*sqrt(Pi)*erf(1/2) - 8/exp(1/4) - 1/3, where erf is the error function. (End)

A221161 G.f.: Sum_{n>=0} (4n+3)^n x^n / (1 + (4n+3)x)^n.

Original entry on oeis.org

1, 7, 72, 1056, 19968, 460800, 12533760, 392232960, 13872660480, 546979184640, 23781703680000, 1130106558873600, 58263271479705600, 3238634262940876800, 193064390900475494400, 12285915784575713280000, 831229959367865401344000, 59578968979556190388224000

Offset: 0

Paul D. Hanna, Jan 03 2013

nonn

			G.f.: A(x) = 1 + 7*x + 72*x^2 + 1056*x^3 + 19968*x^4 + 460800*x^5 +...
where
A(x) = 1 + 7*x/(1+7*x) + 11^2*x^2/(1+11*x)^2 + 15^3*x^3/(1+15*x)^3 + 19^4*x^4/(1+19*x)^4 + 23^5*x^5/(1+23*x)^5 +...

Cf. A187735, A014479, A187738, A187739, A221160, A187740, A333419.

a[n_] := (2*n + 5)*4^(n - 1)*n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Dec 23 2022 *)

{a(n)=polcoeff(sum(m=0,n,((4*m+3)*x)^m/(1+(4*m+3)*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) = (2*n+5) * 4^(n-1) * n!  for n>0 with a(0)=1.
E.g.f.: (1 - x - 4*x^2) / (1-4*x)^2.
From Amiram Eldar, Dec 23 2022: (Start)
Sum_{n>=0} 1/a(n) = 48*sqrt(Pi)*erfi(1/2) - 40*exp(1/4) + 1/5, where erfi is the imaginary error function.
Sum_{n>=0} (-1)^n/a(n) = 48*sqrt(Pi)*erf(1/2) - 56/exp(1/4) + 1/5, where erf is the error function. (End)

A187742 G.f.: Sum_{n>=0} (n+x)^n * x^n / (1 + n*x + x^2)^n.

Original entry on oeis.org

1, 1, 4, 14, 66, 384, 2640, 20880, 186480, 1854720, 20321280, 243129600, 3153427200, 44068147200, 660064204800, 10548573235200, 179151388416000, 3222109642752000, 61178237632512000, 1222853377794048000, 25667116186263552000, 564433265896980480000, 12977099311614197760000

Offset: 0

Paul D. Hanna, Jan 03 2013

nonn

For values of n between 3 and 11 (possibly continuing) the number of conjugacy classes of the symmetric group S_n when conjugating by a single transposition. - Attila Egri-Nagy, Aug 15 2014

			G.f.: A(x) = 1 + x + 4*x^2 + 14*x^3 + 66*x^4 + 384*x^5 + 2640*x^6 +...
where
A(x) = 1 + (1+x)*x/(1+x+x^2) + (2+x)^2*x^2/(1+2*x+x^2)^2 + (3+x)^3*x^3/(1+3*x+x^2)^3 + (4+x)^4*x^4/(1+4*x+x^2)^4 + (5+x)^5*x^5/(1+5*x+x^2)^5 +...

Cf. A202365, A187741, A187735, A187746.

List([3..11], n->Size(OrbitsDomain(Group((1,2)),SymmetricGroup(IsPermGroup, n), \^))); # Attila Egri-Nagy, Aug 15 2014

a[0] = 1; a[1] = 1; a[n_] := (n^2 + n + 2)*(n - 1)!/2; Table[a[n], {n, 0, 20}] (* Wesley Ivan Hurt, Aug 15 2014 *)

{a(n)=polcoeff( sum(m=0, n, (m+x)^m*x^m/(1+m*x+x^2 +x*O(x^n))^m), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=if(n>=0&n<=1,1,(n^2+n+2)*(n-1)!/2)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=n!*polcoeff(1/2 + 1/(2*(1-x)^2) - x - log(1-x +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

x='x+O('x^66); concat([1], Vec(serlaplace(1/(1-x)^3 + x/(1-x)))) \\ Joerg Arndt, Aug 15 2014

a(n) = (n^2+n+2) * (n-1)!/2, for n>1 with a(0)=a(1)=1.
E.g.f.: 1/2 + 1/(2*(1-x)^2) - x - log(1-x).
E.g.f.: Sum_{n>=0} a(n+1)*x^n/n! = 1/(1-x)^3 + x/(1-x).

A229039 G.f.: Sum_{n>=0} (n+2)^n * x^n / (1 + (n+2)*x)^n.

Original entry on oeis.org

1, 3, 7, 24, 108, 600, 3960, 30240, 262080, 2540160, 27216000, 319334400, 4071513600, 56043187200, 828193766400, 13076743680000, 219689293824000, 3912561709056000, 73627297615872000, 1459741204905984000, 30411275102208000000, 664182248232222720000

Offset: 0

Paul D. Hanna, Sep 11 2013

nonn

More generally, we have the identity:
if Sum_{n>=0} a(n)*x^n = Sum_{n>=0} (b*n+c)^n * x^n / (1 + (b*n+c)*x)^n,
then Sum_{n>=0} a(n)*x^n/n! = (2 - 2*(b-c)*x + b*(b-2*c)*x^2)/(2*(1-b*x)^2)
so that a(n) = (b*n + (b+2*c)) * b^(n-1) * n!/2 for n>0 with a(0)=1.

			O.g.f.: A(x) = 1 + 3*x + 7*x^2 + 24*x^3 + 108*x^4 + 600*x^5 + 3960*x^6 +...
where
A(x) = 1 + 3*x/(1+3*x) + 4^2*x^2/(1+4*x)^2 + 5^3*x^3/(1+5*x)^3 + 6^4*x^4/(1+6*x)^4 + 7^5*x^5/(1+7*x)^5 +...
E.g.f.: E(x) = 1 + 3*x + 7*x^2/2! + 24*x^3/3! + 108*x^4/4! + 600*x^5/5! +...
where
E(x) = 1 + 3*x + 7/2*x^2 + 4*x^3 + 9/2*x^4 + 5*x^5 + 11/2*x^6 + 6*x^7 +...
which is the expansion of: (2 + 2*x - 3*x^2) / (2 - 4*x + 2*x^2).

Cf. A038720, A230056, A187735, A187738, A187739, A229039, A221160, A221161, A187740.

a[n_] := (n + 5)*n!/2; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Dec 11 2022 *)

{a(n)=polcoeff( sum(m=0, n, ((m+2)*x)^m / (1 + (m+2)*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=if(n==0,1, (n+5) * n!/2 )}
for(n=0, 20, print1(a(n), ", "))

a(n) = (n+5) * n!/2 for n>0 with a(0)=1.
E.g.f.: (2 + 2*x - 3*x^2)/(2*(1-x)^2).
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 18*e - 237/5.
Sum_{n>=0} (-1)^n/a(n) = 243/5 - 130/e. (End)

A187746 G.f.: Sum_{n>=0} (2n+x)^n x^n / (1 + 2nx + x^2)^n.

Original entry on oeis.org

1, 2, 13, 100, 984, 11712, 163200, 2603520, 46771200, 934133760, 20530298880, 492355584000, 12793813401600, 358063276032000, 10737974299852800, 343513154086502400, 11676590580695040000, 420271561157640192000, 15967576932074127360000

Offset: 0

Paul D. Hanna, Jan 03 2013

nonn

			G.f.: A(x) = 1 + 2*x + 13*x^2 + 100*x^3 + 984*x^4 + 11712*x^5 +...
where
A(x) = 1 + (2+x)*x/(1+2*x+x^2) + (4+x)^2*x^2/(1+4*x+x^2)^2 + (6+x)^3*x^3/(1+6*x+x^2)^3 + (8+x)^4*x^4/(1+8*x+x^2)^4 + (10+x)^5*x^5/(1+10*x+x^2)^5 +...

Cf. A187742, A187735.

{a(n)=polcoeff( sum(m=0, n, (2*m+x)^m*x^m/(1+2*m*x+x^2 +x*O(x^n))^m), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=if(n==0,1,if(n==1,2,(2*n^2+2*n+1)*2^(n-2)*(n-1)!))}
for(n=0, 30, print1(a(n), ", "))

{a(n)=n!*polcoeff(1/2 + 1/(2*(1-2*x)^2) - x/2 - log(1-2*x +x*O(x^n))/4, n)}
for(n=0, 30, print1(a(n), ", "))

a(n) = (2*n^2+2*n+1) * 2^(n-2) * (n-1)! for n>1 with a(0)=1, a(1)=2.
E.g.f.: 1/2 + 1/(2*(1-2*x)^2) - x/2 - log(1-2*x)/4.
E.g.f.: Sum_{n>=0} a(n+1)*x^n/n! = 2/(1-2*x)^3 + x/(1-2*x).

cp's OEIS Frontend

A014479 Exponential generating function = (1+2*x)/(1-2*x)^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A187738 G.f.: Sum_{n>=0} (3*n+1)^n * x^n / (1 + (3*n+1)*x)^n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A187739 G.f.: Sum_{n>=0} (3*n+2)^n * x^n / (1 + (3*n+2)*x)^n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A187740 G.f.: Sum_{n>=0} (5*n+1)^n * x^n / (1 + (5*n+1)*x)^n.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A187741 G.f.: Sum_{n>=0} (1 + n*x)^n * x^n / (1 + x + n*x^2)^n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A221160 G.f.: Sum_{n>=0} (4*n+1)^n * x^n / (1 + (4*n+1)*x)^n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A221161 G.f.: Sum_{n>=0} (4*n+3)^n * x^n / (1 + (4*n+3)*x)^n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A187742 G.f.: Sum_{n>=0} (n+x)^n * x^n / (1 + n*x + x^2)^n.

Views

Author

Keywords

Comments

Examples

A014479 Exponential generating function = (1+2x)/(1-2x)^3.

A187738 G.f.: Sum_{n>=0} (3n+1)^n x^n / (1 + (3n+1)x)^n.

A187739 G.f.: Sum_{n>=0} (3n+2)^n x^n / (1 + (3n+2)x)^n.

A187740 G.f.: Sum_{n>=0} (5n+1)^n x^n / (1 + (5n+1)x)^n.

A187741 G.f.: Sum_{n>=0} (1 + nx)^n x^n / (1 + x + n*x^2)^n.

A221160 G.f.: Sum_{n>=0} (4n+1)^n x^n / (1 + (4n+1)x)^n.

A221161 G.f.: Sum_{n>=0} (4n+3)^n x^n / (1 + (4n+3)x)^n.

A187746 G.f.: Sum_{n>=0} (2n+x)^n x^n / (1 + 2nx + x^2)^n.