A323620 Expansion of e.g.f. 2*sqrt(1 + x)*sinh(sqrt(5)*log(1 + x)/2)/sqrt(5).

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

Offset: 0

Author

Ilya Gutkovskiy, Jan 20 2019

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A000045, A048994, A178840, A213593, A263576, A265165.

Programs

Magma

[(&+[StirlingFirst(n,k)*Fibonacci(k): k in [0..n]]): n in [0..25]]; // G. C. Greubel, Feb 07 2019

Mathematica

FullSimplify[nmax = 23; CoefficientList[Series[2 Sqrt[1 + x] Sinh[Sqrt[5] Log[1 + x]/2]/Sqrt[5], {x, 0, nmax}], x] Range[0, nmax]!]
Table[Sum[StirlingS1[n, k] Fibonacci[k], {k, 0, n}], {n, 0, 23}]

PARI

{a(n) = sum(k=0,n, stirling(n,k,1)*fibonacci(k))};
vector(25, n, n--; a(n)) \\ G. C. Greubel, Feb 07 2019

Sage

[sum((-1)^(k+n)*stirling_number1(n,k)*fibonacci(k) for k in (0..n)) for n in (0..25)] # G. C. Greubel, Feb 07 2019

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000045(k).

From Vaclav Kotesovec, Jan 21 2019: (Start)

a(n) = -(-1)^n * cos(sqrt(5)*Pi/2) * (Gamma((3 + sqrt(5))/2) * Gamma(n - (1 + sqrt(5))/2) - Gamma((3 - sqrt(5))/2) * Gamma(n + (sqrt(5) - 1)/2)) / (Pi*sqrt(5)).

a(n) ~ -(-1)^n * n! / (sqrt(5) * Gamma((sqrt(5)-1)/2) * n^((3 - sqrt(5))/2)).

a(n) = -2*(n-2)*a(n-1) - (n^2 - 5*n + 5)*a(n-2). (End)