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A082139 A transform of binomial(n,5). 11
1, 6, 42, 224, 1008, 4032, 14784, 50688, 164736, 512512, 1537536, 4472832, 12673024, 35094528, 95256576, 254017536, 666796032, 1725825024, 4410441728, 11142168576, 27855421440, 68975329280, 169303080960, 412216197120 (list; graph; refs; listen; history; text; internal format)
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

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) x='x+O('x^30); Vec(serlaplace(x^5*exp(x)*cosh(x)/5!)) \\ G. C. Greubel, Feb 05 2018

CROSSREFS

Cf. A080951, A082138.

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

Sequence in context: A047663 A054642 A321250 * A180286 A054613 A270239

Adjacent sequences:  A082136 A082137 A082138 * A082140 A082141 A082142

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Apr 06 2003

STATUS

approved

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Last modified June 17 15:07 EDT 2019. Contains 324185 sequences. (Running on oeis4.)