A030440 Values of Newton-Gregory forward interpolating polynomial (1/3)*(n-1)*(2*n+3)*(2*n-1).

1, 0, 7, 30, 77, 156, 275, 442, 665, 952, 1311, 1750, 2277, 2900, 3627, 4466, 5425, 6512, 7735, 9102, 10621, 12300, 14147, 16170, 18377, 20776, 23375, 26182, 29205, 32452, 35931, 39650, 43617, 47840, 52327, 57086, 62125, 67452, 73075, 79002, 85241, 91800, 98687, 105910

Offset: 0

Author

Ilias.Kotsireas(AT)lip6.fr (Ilias Kotsireas)

Keywords

• nonn
• easy

Comments

Starting at a(2)=7, partial sums of A073577. - J. M. Bergot, Apr 20 2016

References

• S. Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234; http://www.scirp.org/journal/am; http://dx.doi.org/10.4236/am.2014.515216

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A073577, A000384 (trinomial k=2 column), A106734, A127672, A027907.

Programs

Magma

[(1/3)*(n-1)*(2*n-1)*(2*n+3):n in [0..50]]; // Vincenzo Librandi, Apr 20 2018

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {1, 0, 7, 30}, 40] (* Vincenzo Librandi, Apr 20 2018 *)

PARI

a(n) = (n-1)*(2*n-1)*(2*n+3)/3; \\ Altug Alkan, Apr 19 2018

Formula

G.f.: (1+13*x^2-2*x^3-4*x)/(1-x)^4. - R. J. Mathar, May 18 2014

a(n) = (1/6) * (A106734(2n) - 1), n > 0. - Mathew Englander, Jun 06 2014

E.g.f.: (3 - 3*x + 12*x^2 + 4*x^3)*exp(x)/3. - Ilya Gutkovskiy, Apr 20 2016

a(n+1) = trinomial(2*n+1, 3) = binomial(2*n+1, 3) + (2*n+1)*(2*n) = n*(2*n+1)*(2*n+5)/3, for n >= 0, with the trinomial irregular triangle A027907. a(n+1) = trinomial(2*n+1, 4*n-1), for n >= 1 (symmetry). a(n+1) = Integral_{x=0..2} (1/sqrt(4 - x^2))*(x^2 - 1)^(2*n+1)*R(4*(n-1), x)/Pi with the R polynomial coefficients given in A127672. [Comtet, p. 77, the integral formula for q=3, n -> 2*n+1, k = 3, rewritten with x = 2*cos(phi)]. The g.f. of {a(n+1)}{n>= 0} is x*(7 + 2*x - x^2)/(1 - x)^4. - _Wolfdieter Lang, Apr 19 2018