A277361 a(n) = Sum_{k=0..n} k^3 * binomial(n-k, k).

0, 0, 1, 2, 11, 28, 80, 194, 461, 1036, 2263, 4800, 9960, 20268, 40577, 80086, 156115, 301004, 574744, 1087918, 2043229, 3810320, 7060079, 13004832, 23826480, 43437240, 78827425, 142446698, 256400411, 459826588, 821834624, 1464149690, 2600662157, 4606368148

Offset: 0

Author

Vladimir Reshetnikov, Oct 10 2016

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,5,8,-2,-4,-1).

Crossrefs

Cf. A000032, A000045, A224227.

Programs

Mathematica

Table[(n (3 n^2 + 3 n - 2) LucasL[n] - (5 n^3 + 3 n^2 - 2 n - 2) Fibonacci[n])/50, {n, 0, 30}] (* or *)
Table[Sum[k^3 Binomial[n - k, k], {k, 0, n}], {n, 0, 30}]

PARI

a(n) = sum(k=0, n, k^3*binomial(n-k, k)); \\ Michel Marcus, Oct 11 2016

PARI

a(n) = my(f1=fibonacci(n-1),f2=fibonacci(n)); (n*(3*n^2 + 3*n - 2)*(2*f1+f2) - (5*n^3 + 3*n^2 - 2*n - 2)*f2)/50 \\ Charles R Greathouse IV, Oct 13 2016

Formula

a(n) = (n*(3*n^2 + 3*n - 2)*A000032(n) - (5*n^3 + 3*n^2 - 2*n - 2)*A000045(n))/50.

a(n) ~ (phi^(n-2)*n^3)/25, phi = (1+sqrt(5))/2.

G.f.: x^2*(1 + x*(x-2)*(x-1)^2)/(x^2 + x - 1)^4.

D-finite with recurrence: (n^5 + 3*n^4 - 5*n^3 - 5*n^2 - n + 2)*a(n) + (n+1)^2*(n^3 + 4*n^2 + n - 1)*a(n-1) = n*(n^4 - 5*n^2 + 5*n - 1)*a(n+1).

E.g.f.: exp(x/2)*(5*x*(x^2 + 9*x - 1)*cosh(sqrt(5)*x/2) + sqrt(5)*(5*x^3 + 3*x^2 + 7*x + 2)*sinh(sqrt(5)*x/2))/125. - Stefano Spezia, Mar 20 2023