A254463 a(n) = 15*2^n + 6*4^n + 10*3^n + 3*5^n + 6^n + 21.

56, 126, 378, 1386, 5778, 26226, 126378, 636426, 3314178, 17714466, 96660378, 536249466, 3015243378, 17141522706, 98333399178, 568324150506, 3305074833378, 19319850386946, 113420243462778, 668241096915546, 3948892688324178, 23393955029043186, 138880128205091178

Offset: 0

Author

Luciano Ancora, Jan 31 2015

Keywords

• nonn
• easy

Comments

This is the sequence of sixth terms of "third partial sums of m-th powers".

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Luciano Ancora, Demonstration of formulas, page 2.
• Luciano Ancora, Recurrence relations for partial sums of m-th powers.
• Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Crossrefs

Cf. A062709, A254362, A254363, A254364, A254464.

Programs

Mathematica

Table[15 2^n + 6 4^n + 10 3^n + 3 5^n + 6^n + 21, {n, 0, 25}] (* Michael De Vlieger, Jan 31 2015 *)

PARI

vector(30, n, n--; 15*2^n + 6*4^n + 10*3^n + 3*5^n + 6^n + 21) \\ Colin Barker, Jan 31 2015

Formula

From Colin Barker, Jan 31 2015: (Start)

G.f.: -2*(12276*x^5 - 20578*x^4 + 12831*x^3 - 3766*x^2 + 525*x - 28)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)).

a(n) = 21*a(n-1) - 175*a(n-2) + 735*a(n-3) - 1624*a(n-4) + 1764*a(n-5) - 720*a(n-6). (End)

E.g.f.: exp(x)*(exp(5*x) + 3*exp(4*x) + 6*exp(3*x) + 10*exp(2*x) + 15*exp(x) + 21). - Elmo R. Oliveira, Sep 16 2024