A082139 A transform of binomial(n,5).

1, 6, 42, 224, 1008, 4032, 14784, 50688, 164736, 512512, 1537536, 4472832, 12673024, 35094528, 95256576, 254017536, 666796032, 1725825024, 4410441728, 11142168576, 27855421440, 68975329280, 169303080960, 412216197120

Offset

0,2

Comments

Sixth row of number array A082137. C(n,5) has e.g.f. (x^5/5!)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 (12,-60,160,-240,192,-64).

Formula

Equals 2 * A080952.

a(n) = (2^(n-1) + 0^n/2)*C(n+5, n).

a(n) = Sum_{j=0..n} C(n+5, j+5)*C(j+5, 5)*(1+(-1)^j)/2.

G.f.: (1 -6*x +30*x^2 -80*x^3 +120*x^4 -96*x^5 +32*x^6)/(1-2*x)^6.

E.g.f.: x^5*exp(x)*cosh(x)/5! (preceded by 5 zeros).

a(n) = ceiling(binomial(n+5,5)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006

From Amiram Eldar, Jan 07 2022: (Start)

Sum_{n>=0} 1/a(n) = 20*log(2) - 38/3.

Sum_{n>=0} (-1)^n/a(n) = 1620*log(3/2) - 656. (End)

Example

a(0) = (2^(-1) + 0^0/2)*binomial(5,0) = 2*(1/2) = 1 (use 0^0 = 1).

Maple

[seq (ceil(binomial(n+5, 5)*2^(n-1)), n=0..23)]; # Zerinvary Lajos, Nov 01 2006

Mathematica

Drop[With[{nmax = 56}, CoefficientList[Series[x^5*Exp[x]*Cosh[x]/5!, {x, 0, nmax}], x]*Range[0, nmax]!], 5] (* or *) Join[{1}, Table[2^(n-1)* Binomial[n+5, n], {n, 1, 30}] (* G. C. Greubel, Feb 05 2018 *)

Prog

(Magma) [(Ceiling(Binomial(n+5, 5)*2^(n-1))) : n in [0..30]]; // Vincenzo Librandi, Sep 22 2011
(PARI) my(x='x+O('x^30)); Vec(serlaplace(x^5*exp(x)*cosh(x)/5!)) \\ G. C. Greubel, Feb 05 2018

Crossrefs

Cf. A080951, A080952, A082138.

Cf. A082140, A082141, A082138, A080951, A080929, A057711.

Sequence in context: A047663 A326744 A054642 * A180286 A054613 A270239

Adjacent sequences: A082136 A082137 A082138 * A082140 A082141 A082142

Keyword

easy,nonn

Author

Paul Barry, Apr 06 2003

Status

approved