OFFSET
0,2
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.
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).
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)
EXAMPLE
a(0) = (2^(-1) + 0^0/2)*C(7,0) = 2*(1/2) = 1 (using 0^0=1).
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 *)
PROG
(PARI) my(x='x+O('x^30)); Vec(serlaplace(x^7*exp(x)*cosh(x)/7!)) \\ G. C. Greubel, Feb 05 2018
(Magma) [(2^(n-1) + 0^n/2)*Binomial(n+7, n): n in [0..30]]; // G. C. Greubel, Feb 05 2018
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 06 2003
STATUS
approved