cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A187735 G.f.: Sum_{n>=0} (2*n+1)^n * x^n / (1 + (2*n+1)*x)^n.

Original entry on oeis.org

1, 3, 16, 120, 1152, 13440, 184320, 2903040, 51609600, 1021870080, 22295347200, 531372441600, 13733933875200, 382588157952000, 11426632984166400, 364223926370304000, 12340763622899712000, 442896294466289664000, 16783438527143608320000
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2013

Keywords

Comments

Compare g.f. to the identity (cf. A001710):
Sum_{n>=0} n^n * x^n / (1 + n*x)^n = 1 + Sum_{n>=1} (n+1)!/2 * x^n.
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.

Examples

			G.f.: A(x) = 1 + 3*x + 16*x^2 + 120*x^3 + 1152*x^4 + 13440*x^5 +...
where
A(x) = 1 + 3*x/(1+3*x) + 5^2*x^2/(1+5*x)^2 + 7^3*x^3/(1+7*x)^3 + 9^4*x^4/(1+9*x)^4 + 11^5*x^5/(1+11*x)^5 +...

Crossrefs

Cf. A001710, A014479, A187738, A187739, A221160, A221161, A187740.

Programs

  • Mathematica
    a[n_] := (n + 2)*2^(n - 1)*n!; Array[a, 20, 0] (* Amiram Eldar, Dec 23 2022 *)
  • PARI
    {a(n)=polcoeff( sum(m=0,n,((2*m+1)*x)^m / (1 + (2*m+1)*x +x*O(x^n))^m),n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n) = (n+2)*2^(n-1)*n!}

Formula

a(n) = (n+2) * 2^(n-1) * n!.
E.g.f.: (1-x)/(1-2*x)^2.
From Amiram Eldar, Dec 23 2022: (Start)
Sum_{n>=0} 1/a(n) = 8 - 4*sqrt(e).
Sum_{n>=0} (-1)^n/a(n) = 8 - 12/sqrt(e). (End)

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

Views

Author

Paul D. Hanna, Jan 03 2013

Keywords

Comments

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.

Examples

			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 +...

Crossrefs

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

Programs

  • PARI
    {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), ", "))

Formula

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} (5*n+1)^n * x^n / (1 + (5*n+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

Views

Author

Paul D. Hanna, Jan 03 2013

Keywords

Examples

			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 +...

Crossrefs

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

Programs

  • PARI
    {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), ", "))

Formula

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).

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

Views

Author

Paul D. Hanna, Jan 03 2013

Keywords

Examples

			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 +...

Crossrefs

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

Programs

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

Formula

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} (4*n+3)^n * x^n / (1 + (4*n+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

Views

Author

Paul D. Hanna, Jan 03 2013

Keywords

Examples

			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 +...

Crossrefs

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

Programs

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

Formula

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)

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

Views

Author

Paul D. Hanna, Sep 11 2013

Keywords

Comments

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.

Examples

			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).

Crossrefs

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

Programs

  • Mathematica
    a[n_] := (n + 5)*n!/2; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Dec 11 2022 *)
  • PARI
    {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), ", "))
  • PARI
    {a(n)=if(n==0,1, (n+5) * n!/2 )}
for(n=0, 20, print1(a(n), ", "))

Formula

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)

A230056 G.f.: Sum_{n>=0} (n+3)^n * x^n / (1 + (n+3)*x)^n.

Original entry on oeis.org

1, 4, 9, 30, 132, 720, 4680, 35280, 302400, 2903040, 30844800, 359251200, 4550515200, 62270208000, 915372057600, 14384418048000, 240612083712000, 4268249137152000, 80029671321600000, 1581386305314816000, 32844177110384640000, 715273190403932160000, 16298010552775311360000
Offset: 0

Views

Author

Paul D. Hanna, Oct 07 2013

Keywords

Examples

			O.g.f.: A(x) = 1 + 4*x + 9*x^2 + 30*x^3 + 132*x^4 + 720*x^5 + 4680*x^6 +...
where
A(x) = 1 + 4*x/(1+4*x) + 5^2*x^2/(1+5*x)^2 + 6^3*x^3/(1+6*x)^3 + 7^4*x^4/(1+7*x)^4 + 8^5*x^5/(1+8*x)^5 +...
E.g.f.: E(x) = 1 + 4*x + 9*x^2/2! + 30*x^3/3! + 132*x^4/4! + 720*x^5/5! +...
where
E(x) = 1 + 4*x + 9/2*x^2 + 5*x^3 + 11/2*x^4 + 6*x^5 + 13/2*x^6 + 7*x^7 +...
which is the expansion of: (2 + 4*x - 5*x^2) / (2 - 4*x + 2*x^2).

Crossrefs

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

Programs

  • Maple
    a:=series(add((n+3)^n*x^n/(1+(n+3)*x)^n,n=0..100),x=0,23): seq(coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    a[n_] := (n + 7)*n!/2; a[0] = 1; Array[a, 25, 0] (* Amiram Eldar, Dec 11 2022 *)
  • PARI
    {a(n)=polcoeff( sum(m=0, n, ((m+3)*x)^m / (1 + (m+3)*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=if(n==0, 1, (n+7) * n!/2 )}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = (n+7) * n!/2 for n>0 with a(0)=1.
E.g.f.: (2 + 4*x - 5*x^2)/(2*(1-x)^2).
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 530*e - 10075/7.
Sum_{n>=0} (-1)^n/a(n) = 10085/7 - 3914/e. (End)

A229036 G.f.: Sum_{n>=0} (3*n-1)^n * x^n / (1 + (3*n-1)*x)^n.

Original entry on oeis.org

1, 2, 21, 270, 4212, 77760, 1662120, 40415760, 1102248000, 33331979520, 1107097891200, 40069801094400, 1569793384051200, 66185883219456000, 2988292627358438400, 143855017177487616000, 7355369573944584192000, 398090614491857903616000, 22737098558477268725760000
Offset: 0

Views

Author

Paul D. Hanna, Sep 11 2013

Keywords

Comments

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.

Examples

			O.g.f.: A(x) = 1 + 2*x + 21*x^2 + 270*x^3 + 4212*x^4 + 77760*x^5 +...
where
A(x) = 1 + 2*x/(1+2*x) + 5^2*x^2/(1+5*x)^2 + 8^3*x^3/(1+8*x)^3 + 11^4*x^4/(1+11*x)^4 + 14^5*x^5/(1+14*x)^5 +...
E.g.f.: E(x) = 1 + 2*x + 21*x^2/2! + 270*x^3/3! + 4212*x^4/4! + 77760*x^5/5! +...
where
E(x) =  1 + 2*x + 21/2*x^2 + 45*x^3 + 351/2*x^4 + 648*x^5 + 4617/2*x^6 +...
which is the expansion of: (2 - 8*x + 15*x^2) / (2 - 12*x + 18*x^2).

Crossrefs

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

Programs

  • Mathematica
    Join[{1},Table[(3n+1)3^(n-1) n!/2,{n,20}]] (* Harvey P. Dale, Feb 10 2015 *)
  • PARI
    {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), ", "))
  • PARI
    {a(n) = if(n==0,1,(3*n+1)*3^(n-1)*n!/2)}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) = (3*n+1) * 3^(n-1) * n!/2 for n>0 with a(0)=1.
E.g.f.: (2 - 8*x + 15*x^2)/(2*(1-3*x)^2).
Showing 1-8 of 8 results.

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