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 1-6 of 6 results.

A227681 G.f.: exp( Sum_{n>=1} x^n * (1+x)^n / (n*(1-x^n)) ).

Original entry on oeis.org

1, 1, 3, 6, 12, 23, 43, 79, 142, 252, 442, 766, 1316, 2244, 3799, 6393, 10704, 17841, 29618, 49000, 80823, 132964, 218242, 357501, 584608, 954553, 1556575, 2535425, 4125805, 6708143, 10898897, 17696749, 28719276, 46586050, 75538702, 122444483, 198420445, 321461918
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 12*x^4 + 23*x^5 + 43*x^6 + 79*x^7 +...
where
log(A(x)) = x*(1+x)/(1-x) + x^2*(1+x)^2/(2*(1-x^2)) + x^3*(1+x)^3/(3*(1-x^3)) + x^4*(1+x)^4/(4*(1-x^4)) + x^5*(1+x)^5/(5*(1-x^5)) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 10*x^3/3 + 17*x^4/4 + 26*x^5/5 + 38*x^6/6 + 57*x^7/7 + 81*x^8/8 + 118*x^9/9 + 180*x^10/10 +...

Links

Crossrefs

Cf. A160571, A227682, A306565, A306749.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/(1 - x^k*(1 + x)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 31 2021 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*(1+x)^m/(1-x^m +x*O(x^n)) )), n)}
for(n=0, 50, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, x^m*sumdiv(m, d, (1+x +x*O(x^n))^d/d) )), n)}
for(n=0, 50, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} x^n * Sum_{d|n} (1+x)^d / d ).
G.f.: Product {n >= 1} 1/(1 - (1 + x)*x^n). - Peter Bala, Jan 20 2015
a(n) ~ phi^(n+1) / (sqrt(5)* A276987), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 16 2019

A123741 A second version of Fibonacci factorials besides A003266.

Original entry on oeis.org

1, 2, 24, 630, 52800, 11381760, 6738443712, 10487895163200, 43294107630090240, 469590163875486482400, 13388418681612808458240000, 1001088091286168023193223168000, 196239953628635168336022309340569600
Offset: 1

Views

Author

Wolfdieter Lang, Oct 13 2006

Keywords

Comments

The formula below is a generalization of n! = Product_{j=1..n} ((n+1) - j) with numbers k replaced by Fibonacci numbers F(k+1):=A000045(k+1), k>=1.
These numbers come up in Vandermonde determinants involving Fibonacci numbers [F(2),...,F(n+1)]. See A123742.

Examples

			n=3: (5-1)*(5-2)*(5-3) = 4*3*2 = 24;
n=4: (8-1)*(8-2)*(8-3)*(8-5) = 7*6*5*3 = 630.

Links

Crossrefs

Cf. A003266 (the usual Fibonacci factorials), A123742.

Programs

  • GAP
    F:=Fibonacci;; List([1..20], n-> Product([1..n], j-> F(n+2) - F(j+1))); # G. C. Greubel, Aug 10 2019
  • Magma
    F:=Fibonacci; [(&*[F(n+2)-F(j+1): j in [1..n]]): n in [1..20]] // G. C. Greubel, Aug 10 2019
  • Maple
    with(combinat): seq(mul(fibonacci(n+2)-fibonacci(j+1), j = 1..n), n = 1 .. 20); # G. C. Greubel, Aug 10 2019
  • Mathematica
    With[{F=Fibonacci}, Table[Product[F[n+2]-F[j+1],{j,n}], {n,20}]] (* G. C. Greubel, Aug 10 2019 *)
  • PARI
    vector(20, n, f=fibonacci; prod(j=1,n, f(n+2)-f(j+1))) \\ G. C. Greubel, Aug 10 2019
  • Sage
    f=fibonacci; [prod(f(n+2)-f(j+1) for j in (1..n)) for n in (1..20)] # G. C. Greubel, Aug 10 2019

Formula

a(n) = Product_{j=1..n} (F(n+2) - F(j+1)), n>=1.
a(n) ~ c * phi^(n*(n+2)) / 5^(n/2), where c = A276987 = QPochhammer(1/phi) and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 31 2021

A274983 a(n) = [n]phi! + [n]{1-phi}!, where [n]_q! is the q-factorial, phi = (1+sqrt(5))/2.

Original entry on oeis.org

2, 2, 3, 14, 130, 2120, 58120, 2636360, 196132320, 23805331920, 4698862837680, 1505416321070640, 781888977967152000, 657866357975539785600, 896265744457831561756800, 1976607903479486428467148800, 7055269158071576119808840371200
Offset: 0

Views

Author

Vladimir Reshetnikov, Sep 23 2016

Keywords

Examples

			For n = 3, [3]_phi! = 1060 + 474*sqrt(5), so a(5) = 2*1060 = 2120 and A274985(5) = 2*474 = 948.

Links

Crossrefs

Cf. A274985, A005329, A275706, A276474, A276688, A276987.

Programs

  • Mathematica
    Round@Table[QFactorial[n, GoldenRatio] + QFactorial[n, 1 - GoldenRatio], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)

Formula

a(n) ~ c * phi^(n*(n+3)/2), where c = QPochhammer(phi-1) = A276987 = 0.1208019218617061294237231569887920563043992516794... . - Vaclav Kotesovec, Sep 24 2016
From Vladimir Reshetnikov, Sep 24 2016 (Start)
[n]_phi! = (a(n) + A274985(n)*sqrt(5))/2.
[n]_{1-phi}! = (a(n) - A274985(n)*sqrt(5))/2. (End)

A274985 a(n) = ([n]phi! - [n]{1-phi}!)/sqrt(5), where [n]_q! is the q-factorial, phi = (1+sqrt(5))/2.

Original entry on oeis.org

0, 0, 1, 6, 58, 948, 25992, 1179016, 87713040, 10646068080, 2101395344400, 673242645670320, 349671381118477440, 294206779308703578240, 400822226102433353285760, 883965927408694948620295680, 3155212287401150653204012531200
Offset: 0

Views

Author

Vladimir Reshetnikov, Sep 23 2016

Keywords

Examples

			For n = 3, [3]_phi! = 1060 + 474*sqrt(5), so A274983(5) = 2*1060 = 2120 and a(5) = 2*474 = 948.

Links

Crossrefs

Cf. A274983, A005329, A275706, A276474, A276688, A276987.

Programs

  • Mathematica
    Round@Table[(QFactorial[n, GoldenRatio] - QFactorial[n, 1 - GoldenRatio])/Sqrt[5], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)

Formula

[n]_phi! = (A274983(n) + a(n)*sqrt(5))/2.
[n]_{1-phi}! = (A274983(n) - a(n)*sqrt(5))/2.
a(n) ~ c * phi^(n*(n+3)/2) / sqrt(5), where c = QPochhammer(phi-1) = A276987 = 0.1208019218617061294237231569887920563043992516794... . - Vaclav Kotesovec, Sep 24 2016

A276990 a(n) = (phi; phi)_n + (1-phi; 1-phi)_n, where (q; q)_n is the q-Pochhammer symbol, phi = (1+sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

2, 1, 2, -2, 20, -190, 3240, -90800, 4174920, -313173840, 38204662320, -7564715117520, 2428250059593600, -1262694691720176000, 1063187432567808662400, -1449125250052431355430400, 3196769645011428154428883200, -11412468527893653264760022630400
Offset: 0

Views

Author

Vladimir Reshetnikov, Sep 24 2016

Keywords

Links

Crossrefs

Cf. A274983, A276474, A276987, A276991.

Programs

  • Mathematica
    Round@Table[QPochhammer[GoldenRatio, GoldenRatio, n] + QPochhammer[1 - GoldenRatio, 1 - GoldenRatio, n], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)

Formula

(phi; phi)_n = (a(n) + A276991(n)*sqrt(5))/2.
(1-phi; 1-phi)_n = (a(n) - A276991(n)*sqrt(5))/2.
a(n) ~ c * (-1)^n * phi^(n*(n+1)/2), where c = (1/phi)_inf = A276987 = 0.1208019218617061294237231569887920563043992516794...

A276991 a(n) = ((phi; phi)_n - (1-phi; 1-phi)_n)/sqrt(5), where (q; q)_n is the q-Pochhammer symbol, phi = (1+sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

0, -1, 0, -2, 8, -86, 1448, -40608, 1867080, -140055600, 17085644400, -3383043446640, 1085946439923840, -564694233102890880, 475471874409018791040, -648068513405723438730240, 1429638846930684965104992000, -5103811083889432701541321459200
Offset: 0

Views

Author

Vladimir Reshetnikov, Sep 24 2016

Keywords

Links

Crossrefs

Cf. A274983, A276474, A276987, A276990.

Programs

  • Mathematica
    Round@Table[(QPochhammer[GoldenRatio, GoldenRatio, n] - QPochhammer[1 - GoldenRatio, 1 - GoldenRatio, n])/Sqrt[5], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)

Formula

(phi; phi)_n = (A276990(n) + a(n)*sqrt(5))/2.
(1-phi; 1-phi)_n = (A276990(n) - a(n)*sqrt(5))/2.
a(n) ~ c * (-1)^n * phi^(n*(n+1)/2) / sqrt(5), where c = (1/phi)_inf = A276987 = 0.1208019218617061294237231569887920563043992516794...
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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