A082138 A transform of C(n,3).

1, 4, 20, 80, 280, 896, 2688, 7680, 21120, 56320, 146432, 372736, 931840, 2293760, 5570560, 13369344, 31752192, 74711040, 174325760, 403701760, 928514048, 2122317824, 4823449600, 10905190400, 24536678400, 54962159616, 122607894528

Offset: 0

Author

Paul Barry, Apr 06 2003

Keywords

• easy
• nonn

Comments

Fourth row of number array A082137. C(n,3) has e.g.f. (x^3/3!)exp(x). The transform averages the binomial and inverse binomial transforms.

Examples

			a(0) = (2^(-1) + 0^0/2)*C(3,0) = 2*(1/2) = 1 (using 0^0=1).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Crossrefs

Cf. A080929, A082137, A082139, A082140, A082141, A080951, A057711.

Programs

GAP

a:=[4,20,80,280];; for n in [5..30] do a[n]:=8*a[n-1]-24*a[n-2] +32*a[n-3]-16*a[n-4]; od; Concatenation([1], a);

Magma

[(Ceiling(Binomial(n+3, 3)*2^(n-1))) : n in [0..30]]; // Vincenzo Librandi, Sep 22 2011

Maple

[seq (ceil(binomial(n+3,3)*2^(n-1)),n=0..30)]; # Zerinvary Lajos, Nov 01 2006

Mathematica

Join[{1}, LinearRecurrence[{8,-24,32,-16}, {4,20,80,280}, 30]] (* G. C. Greubel, Jul 23 2019 *)

PARI

my(x='x+O('x^30)); Vec((1-4*x+12*x^2-16*x^3 + 8*x^4)/(1-2*x)^4) \\ G. C. Greubel, Jul 23 2019

Sage

((1-4*x+12*x^2-16*x^3+8*x^4)/(1-2*x)^4).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 23 2019

Formula

a(n) = (2^(n-1) + 0^n/2)*C(n+3, n).

a(n) = Sum_{j=0..n} C(n+3, j+3)*C(j+3, 3)*(1 + (-1)^j)/2.

G.f.: (1 - 4*x + 12*x^2 - 16*x^3 + 8*x^4)/(1-2*x)^4.

E.g.f.: (x^3/3!)*exp(x)*cosh(x) (preceded by 3 zeros).

a(n) = ceiling(binomial(n+3,3)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006

From Amiram Eldar, Jan 07 2022: (Start)

Sum_{n>=0} 1/a(n) = 12*log(2) - 7.

Sum_{n>=0} (-1)^n/a(n) = 108*log(3/2) - 43. (End)