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 1-2 of 2 results.

A213593 Stirling transform of the first kind of the Lucas numbers A000032.

Original entry on oeis.org

2, 1, 2, -3, 10, -45, 250, -1645, 12490, -107415, 1031690, -10943955, 127058690, -1602400085, 21812913650, -318763741725, 4977247397650, -82695799908975, 1456703469048850, -27117356172328675, 531930264143933050
Offset: 0

Views

Author

Jiaqiang Pan, Jun 15 2012

Keywords

Examples

			For n=4, a(4) = r*(r-1)*(r-2)*(r-3) + s*(s-1)*(s-2)*(s-3) = 10.
		

Crossrefs

Cf. A000032 (Lucas numbers), the general-term formula of which is L(n) = r^n + s^n.

Programs

  • GAP
    Concatenation([2], List([1..25], n-> Sum([1..n], k-> (-1)^(n-k)* Stirling1(n, k)*Lucas(1,-1,k)[2] ))) # G. C. Greubel, Jul 06 2019
  • Magma
    [2] cat [(&+[StirlingFirst(n,k)*Lucas(k): k in [1..n]]): n in [1..25]]; // G. C. Greubel, Jul 06 2019
    
  • Maple
    A000032 := proc(n)
            combinat[fibonacci](n+1)+combinat[fibonacci](n-1) ;
    end proc:
    A213593 := proc(n)
            add(combinat[stirling1](n,i)*A000032(i),i=0..n) ;
    end proc:
    seq(A213593(n),n=0..20) ; # R. J. Mathar, Jun 26 2012
  • Mathematica
    Expand@FunctionExpand@Table[Gamma[2 - GoldenRatio]/Gamma[2 - GoldenRatio - n] + Gamma[1 + GoldenRatio]/Gamma[1 - n + GoldenRatio], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)
    Table[If[n==0, 2, Sum[StirlingS1[n, k]*LucasL[k], {k, n}]], {n, 0, 25}] (* G. C. Greubel, Jul 06 2019 *)
  • PARI
    lucas(n) = fibonacci(n+1) + fibonacci(n-1);
    vector(25, n, n--; if(n==0, 2, sum(k=1,n, stirling(n,k,1)*lucas(k)) )) \\ G. C. Greubel, Jul 06 2019
    
  • Sage
    [2]+[sum((-1)^(n-k)*stirling_number1(n,k)*lucas_number2(k,1,-1) for k in (1..n)) for n in (1..25)] # G. C. Greubel, Jul 06 2019
    

Formula

a(0)=2; for n=1,2,3, ..., a(n) = r*(r-1)*(r-2)*...*(r-n+1) + s*(s-1)*(s-2)*...*(s-n+1), where r=(1+sqrt(5))/2 and s=(1-sqrt(5))/2.
From Vladimir Reshetnikov, Oct 20 2015: (Start)
Let phi=(1+sqrt(5))/2.
a(n) = Gamma(2-phi)/Gamma(2-phi-n)+Gamma(1+phi)/Gamma(1+phi-n).
Recurrence: a(0)=2, a(1)=1, a(n+2) = (1+n-n^2)*a(n) - 2*n*a(n+1).
E.g.f.: (1+(x+1)^sqrt(5))/(x+1)^(1/phi).
(End)
a(n) ~ (-1)^n * n! * n^((sqrt(5)-3)/2) / Gamma(2/(1+sqrt(5))). - Vaclav Kotesovec, Oct 21 2015
a(n) = Sum_{k=1..n} Stirling1(n,k)*Lucas(k). - G. C. Greubel, Jul 06 2019

A306183 The coefficients of x in the reduction of x^2 -> x + 1 for the polynomial p(n,x) = Product_{k=1..n} (x+k).

Original entry on oeis.org

0, 1, 4, 19, 108, 719, 5496, 47465, 457160, 4858865, 56490060, 713165035, 9715762980, 142069257055, 2219386098160, 36889108220305, 650018185589520, 12103669982341025, 237476572759473300, 4896758300881695875, 105866710959427454300, 2394660132226522508975, 56560492065670933962600
Offset: 0

Views

Author

G. C. Greubel, Feb 07 2019

Keywords

Comments

See A192936 for the constant term of the reduction x^2 -> x + 1 for the polynomial p(n,x) = Product_{k=1..n} (x+k).

Crossrefs

Cf. A192936, A323620 (signed)

Programs

  • Magma
    [(-1)^(n+1)*(&+[StirlingFirst(n+2,k)*Fibonacci(k): k in [0..n+2]]): n in [0..30]];
    
  • Mathematica
    Table[(-1)^(n+1)*Sum[StirlingS1[n+2,k]*Fibonacci[k],{k,0,n+2}],{n,0,30}]
  • PARI
    {a(n) = (-1)^(n+1)*sum(k=0,n+2, stirling(n+2,k,1)*fibonacci(k))};
    vector(30, n, n--; a(n))
    
  • Sage
    [sum((-1)^(k+1)*stirling_number1(n+2,k)*fibonacci(k) for k in (0..n+2)) for n in (0..30)]

Formula

a(n) = (-1)^(n+1)*Sum_{k=0..n+2} Stirling1(n+2,k)*A000045(k).
From Vaclav Kotesovec, Feb 09 2019: (Start)
a(n) = 2*n*a(n-1) - (n^2 - n - 1)*a(n-2).
a(n) = cos(Pi*sqrt(5)/2) * (Gamma(sqrt(5)*phi) * Gamma(n + 1/phi^2) / phi^2 - phi^2 * Gamma(sqrt(5)/phi) * Gamma(n + phi^2)) / (Pi*sqrt(5)).
a(n) ~ c * n! * n^phi, where c = -cos(sqrt(5)*Pi/2) * (5 + 3*sqrt(5)) * Gamma((5 - sqrt(5))/2) / (10*Pi) = 0.30858712435869... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. (End)
Showing 1-2 of 2 results.