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).

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

Offset

0,3

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).

Links

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

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)

Mathematica

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

Prog

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

Crossrefs

Cf. A192936, A323620 (signed)

Sequence in context: A206227 A091643 A323620 * A241840 A199318 A117397

Adjacent sequences: A306180 A306181 A306182 * A306184 A306185 A306186

Keyword

nonn

Author

G. C. Greubel, Feb 07 2019

Status

approved