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

0,2

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.

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).

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)

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A295116 A303508 A258627 * A074358 A255050 A371408

Adjacent sequences: A082135 A082136 A082137 * A082139 A082140 A082141

Keyword

easy,nonn

Author

Paul Barry, Apr 06 2003

Status

approved