A323620 -id:A323620 - cp's OEIS frontend

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

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

Offset: 0

Jiaqiang Pan, Jun 15 2012

sign

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

G. C. Greubel, Table of n, a(n) for n = 0..445
Jiaqiang Pan, Matrix Decomposition of the Unified Generalized Stirling Numbers and Inversion of the Generalized Factorial Matrices, Journal of Integer Sequences 15 (2012) #12.6.6

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

Cf. A008275, A323620.

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

[2] cat [(&+[StirlingFirst(n,k)*Lucas(k): k in [1..n]]): n in [1..25]]; // G. C. Greubel, Jul 06 2019

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

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 *)

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

[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

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

G. C. Greubel, Feb 07 2019

nonn

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).

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A192936, A323620 (signed)

[(-1)^(n+1)*(&+[StirlingFirst(n+2,k)*Fibonacci(k): k in [0..n+2]]): n in [0..30]];

Table[(-1)^(n+1)*Sum[StirlingS1[n+2,k]*Fibonacci[k],{k,0,n+2}],{n,0,30}]

{a(n) = (-1)^(n+1)*sum(k=0,n+2, stirling(n+2,k,1)*fibonacci(k))};
vector(30, n, n--; a(n))

[sum((-1)^(k+1)*stirling_number1(n+2,k)*fibonacci(k) for k in (0..n+2)) for n in (0..30)]

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)

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula