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 1-4 of 4 results.

A111139 a(n) = n!*Sum_{k=0..n} Fibonacci(k)/k!.

Original entry on oeis.org

0, 1, 3, 11, 47, 240, 1448, 10149, 81213, 730951, 7309565, 80405304, 964863792, 12543229529, 175605213783, 2634078207355, 42145251318667, 716469272418936, 12896446903543432, 245032491167329389, 4900649823346594545
Offset: 0

Views

Author

Vladeta Jovovic, Oct 17 2005

Keywords

Comments

Eigensequence of a triangle with the Fibonacci series as the left border, the natural numbers (1, 2, 3, ...) as the right border; and the rest zeros. - Gary W. Adamson, Aug 01 2016

Links

Crossrefs

Cf. A000045, A263823.
Cf. A009102, A009551, A000142, A000166, A000522, A000023, A053486, A010844 (incomplete Gamma function values at other points).

Programs

  • Maple
    a:=n->sum(fibonacci (j)*n!/j!,j=0..n):seq(a(n),n=0..20); # Zerinvary Lajos, Mar 19 2007
  • Mathematica
    f[n_] := n!*Sum[Fibonacci[k]/k!, {k, 0, n}]; Table[ f[n], {n, 0, 20}] (* or *)
Simplify[ Range[0, 20]!CoefficientList[ Series[2/Sqrt[5]*Exp[x/2]*Sinh[Sqrt[5]*x/2]/(1 - x), {x, 0, 20}], x]] (* Robert G. Wilson v, Oct 21 2005 *)
Module[{nn=20,fibs,fct},fct=Range[0,nn]!;fibs=Accumulate[ Fibonacci[ Range[ 0,nn]]/fct];Times@@@Thread[{fct,fibs}]] (* Harvey P. Dale, Feb 19 2014 *)
Round@Table[(E^GoldenRatio Gamma[n+1, GoldenRatio] - E^(1-GoldenRatio) Gamma[n+1, 1-GoldenRatio])/Sqrt[5], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 27 2015 *)
  • PARI
    vector(100, n, n--; n!*sum(k=0, n, fibonacci(k)/k!)) \\ Altug Alkan, Oct 28 2015

Formula

E.g.f.: (2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2)/(1-x).
Recurrence: a(n) = (n+1)*a(n-1) - (n-2)*a(n-2) - (n-2)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 2*sqrt(e/5)*sinh(sqrt(5)/2)*n!. - Vaclav Kotesovec, Oct 18 2012
From Vladimir Reshetnikov, Oct 27 2015: (Start)
Let phi=(1+sqrt(5))/2.
a(n) = (phi^n*hypergeom([1,-n], [], 1-phi)-(1-phi)^n*hypergeom([1,-n], [], phi))/sqrt(5).
a(n) = (exp(phi)*Gamma(n+1, phi)-exp^(1-phi)*Gamma(n+1, 1-phi))/sqrt(5), where Gamma(a, x) is the upper incomplete Gamma function.
Gamma(n+1, phi)*exp(phi) = a(n)*phi + A263823(n).
a(n) ~ exp(phi-n)*n^(n+1/2)*sqrt(2*Pi/5)*(1-exp(-sqrt(5))).
(End)

A277345 a(n) = Gamma(n+1, phi)*exp(phi) + Gamma(n+1, 1-phi)*exp(1-phi), where phi=(1+sqrt(5))/2.

Original entry on oeis.org

2, 3, 9, 31, 131, 666, 4014, 28127, 225063, 2025643, 20256553, 222822282, 2673867706, 34760280699, 486643930629, 7299658960799, 116794543374991, 1985507237378418, 35739130272817302, 679043475183538087, 13580869503670776867, 285198259577086338683
Offset: 0

Views

Author

Vladimir Reshetnikov, Oct 09 2016

Keywords

Comments

Gamma(a, x) is the upper incomplete Gamma function.

Links

Crossrefs

Cf. A263823.

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 2, a[1] == 3, a[2] == 9, n (a[n] + a[n - 1]) == (n + 3) a[n + 1] - a[n + 2]}, a[n], {n, 0, 20}] (* or *)
Round@Table[Gamma[n + 1, GoldenRatio] Exp[GoldenRatio] + Gamma[n + 1, 1 - GoldenRatio] Exp[1 - GoldenRatio], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)

Formula

E.g.f: (exp(phi*x) + exp((1-phi)*x))/(1-x).
Recurrence: n*(a(n) + a(n-1)) = (n+3)*a(n+1) - a(n+2).
a(n) ~ 2*exp(1/2)*cosh(sqrt(5)/2) * (n-1)!. - Vaclav Kotesovec, Oct 10 2016

A277431 Expansion of e.g.f.: cosh(sqrt(2)*x)/(1-x).

Original entry on oeis.org

1, 1, 4, 12, 52, 260, 1568, 10976, 87824, 790416, 7904192, 86946112, 1043353408, 13563594304, 189890320384, 2848354805760, 45573676892416, 774752507171072, 13945545129079808, 264965357452516352, 5299307149050328064, 111285450130056889344, 2448279902861251567616
Offset: 0

Views

Author

Vladimir Reshetnikov, Oct 14 2016

Keywords

Examples

			G.f. = 1 + x + 4*x^2 + 12*x^3 + 52*x^4 + 260*x^5 + 1568*x^6 + ... - _Michael Somos_, Oct 01 2018

Links

Crossrefs

Cf. A000023, A263823, A277345, A277432.

Programs

  • Magma
    I:=[1,4,12]; [1] cat [n le 3 select I[n] else n*Self(n-1) + 2*Self(n-2) - 2*(n-2)*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 30 2018
  • Mathematica
    Round@Table[(Gamma[n + 1, Sqrt[2]] Exp[Sqrt[2]] + Gamma[n + 1, -Sqrt[2]]/Exp[Sqrt[2]])/2, {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)
Table[SeriesCoefficient[Cosh[Sqrt[2] x]/(1 - x), {x, 0, n}] n!, {n, 0, 20}]
a[ n_] := If[ n < 0, 0, n! Sum[ 2^k / (2 k)!, {k, 0, n/2}]]; (* Michael Somos, Oct 01 2018 *)
With[{nn=30},CoefficientList[Series[Cosh[x Sqrt[2]]/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 13 2024 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(cosh(sqrt(2)*x)/(1-x))) \\ G. C. Greubel, Sep 30 2018
  • PARI
    {a(n) = if( n<0, 0, n! * sum(k=0, n\2, 2^k / (2*k)!))}; /* Michael Somos, Oct 01 2018 */

Formula

a(n) = (Gamma(n+1, sqrt(2))*exp(sqrt(2)) + Gamma(n+1, -sqrt(2))/exp(sqrt(2)))/2.
a(n) ~ sqrt(2*Pi)*cosh(sqrt(2))*n^(n+1/2)*exp(-n).
D-finite with recurrence: a(n) = n*a(n-1) + 2*a(n-2) - 2*(n-2)*a(n-3).
Gamma(n+1, sqrt(2))*exp(sqrt(2)) = a(n) + sqrt(2)*A277432(n).
Gamma(n+1, -sqrt(2))/exp(sqrt(2)) = a(n) - sqrt(2)*A277432(n).
a(2*n+1) = (2*n+1)*a(2*n).
0 = a(n)*(+4*a(n+1) -4*a(n+2) -6*a(n+3) +2*a(n+4)) +a(n+1)*(+4*a(n+1) +2*a(n+2) -4*a(n+3)) +a(n+2)*(+2*a(n+2) +a(n+3) -a(n+4)) + a(n+3)*(+a(n+3)) for all n>-3. - Michael Somos, Oct 01 2018

A277432 E.g.f.: sinh(sqrt(2)*x)/(sqrt(2)*(1-x)).

Original entry on oeis.org

0, 1, 2, 8, 32, 164, 984, 6896, 55168, 496528, 4965280, 54618112, 655417344, 8520425536, 119285957504, 1789289362688, 28628629803008, 486686706651392, 8760360719725056, 166446853674776576, 3328937073495531520, 69907678543406162944, 1537968927954935584768
Offset: 0

Views

Author

Vladimir Reshetnikov, Oct 14 2016

Keywords

Links

Crossrefs

Cf. A000023, A263823, A277345, A277431.

Programs

  • Magma
    I:=[1,2,8]; [0] cat [n le 3 select I[n] else n*Self(n-1) + 2*Self(n-2) - 2*(n-2)*Self(n-3): n in [1..30]]; // G. C. Greubel, Aug 19 2018
  • Maple
    f:= gfun:-rectoproc({a(n) = n*a(n-1) + 2*a(n-2) - 2*(n-2)*a(n-3),a(0)=0,a(1)=1,a(2)=2},a(n),remember):
map(f, [$0..20]); # Robert Israel, Oct 30 2016
  • Mathematica
    Round@Table[(Gamma[n + 1, Sqrt[2]] Exp[Sqrt[2]] - Gamma[n + 1, -Sqrt[2]]/Exp[Sqrt[2]])/(2 Sqrt[2]), {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)
Expand@Table[SeriesCoefficient[Sinh[Sqrt[2] x]/(Sqrt[2] (1 - x)), {x, 0, n}] n!, {n, 0, 20}]
  • PARI
    x='x+O('x^30); concat([0], round(Vec(serlaplace(sinh(sqrt(2)*x)/( sqrt(2)*(1-x)))))) \\ G. C. Greubel, Aug 19 2018

Formula

a(n) = (Gamma(n+1, sqrt(2))*exp(sqrt(2)) - Gamma(n+1, -sqrt(2))/exp(sqrt(2))) / (2*sqrt(2)).
a(n) ~ sqrt(Pi)*sinh(sqrt(2))*n^(n+1/2)*exp(-n).
D-finite with recurrence: a(n) = n*a(n-1) + 2*a(n-2) - 2*(n-2)*a(n-3).
Gamma(n+1, sqrt(2))*exp(sqrt(2)) = A277431(n) + sqrt(2)*a(n).
Gamma(n+1, -sqrt(2))/exp(sqrt(2)) = A277431(n) - sqrt(2)*a(n).
For n > 0, a(2*n) = 2*n*a(2*n-1).
Showing 1-4 of 4 results.

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