A082141 A transform of C(n,7).

1, 8, 72, 480, 2640, 12672, 54912, 219648, 823680, 2928640, 9957376, 32587776, 103194624, 317521920, 952565760, 2794192896, 8033304576, 22682271744, 63006310400, 172438323200, 465583472640, 1241555927040, 3273192898560

Offset: 0

Author

Paul Barry, Apr 06 2003

Keywords

• easy
• nonn

Comments

Eighth row of number array A082137. C(n,7) has e.g.f. (x^7/7!)exp(x). The transform averages the binomial and inverse binomial transforms.

Examples

			a(0) = (2^(-1) + 0^0/2)*C(7,0) = 2*(1/2) = 1 (using 0^0=1).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792,1024,-256).

Crossrefs

Cf. A054851, A082137, A082139, A082140.

Programs

Magma

[(2^(n-1) + 0^n/2)*Binomial(n+7,n): n in [0..30]]; // G. C. Greubel, Feb 05 2018

Maple

[seq (ceil(binomial(n+7,7)*2^(n-1)),n=0..22)]; # Zerinvary Lajos, Nov 01 2006

Mathematica

Drop[With[{nmax = 50}, CoefficientList[Series[x^7*Exp[x]*Cosh[x]/7!, {x, 0, nmax}], x]*Range[0, nmax]!], 5] (* or *) Join[{1}, Table[2^(n-1)* Binomial[n+7,n], {n,1,30}]] (* G. C. Greubel, Feb 05 2018 *)

PARI

my(x='x+O('x^30)); Vec(serlaplace(x^7*exp(x)*cosh(x)/7!)) \\ G. C. Greubel, Feb 05 2018

Formula

a(n) = (2^(n-1) + 0^n/2)*C(n+7,n).

a(n) = Sum_{j=0..n} C(n+7, j+7)*C(j+7, 7)*(1+(-1)^j)/2.

G.f.: (1 - 8*x + 56*x^2 - 224*x^3 + 560*x^4 - 896*x^5 + 896*x^6 - 512*x^7 + 128*x^8)/(1-2*x)^8.

E.g.f.: (x^7/7!)*exp(x)*cosh(x) (with 7 leading zeros).

a(n) = ceiling(binomial(n+7,7)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006

From Amiram Eldar, Jan 07 2022: (Start)

Sum_{n>=0} 1/a(n) = 28*log(2) - 274/15.

Sum_{n>=0} (-1)^n/a(n) = 20412*log(3/2) - 124132/15. (End)