OFFSET
0,2
COMMENTS
Seventh row of number array A082137. C(n,6) has e.g.f. (x^6/6!)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 (14,-84,280,-560,672,-448,128).
FORMULA
a(n) = (2^(n-1) + 0^n/2)*C(n+6,n).
a(n) = Sum_{j=0..n} C(n+6, j+6)*C(j+6, 6)*(1+(-1)^j)/2.
G.f.: (1 - 7*x + 42*x^2 - 140*x^3 + 280*x^4 - 336*x^5 + 224*x^6 - 64*x^7)/ (1-2*x)^7.
E.g.f.: (x^6/6!)*exp(x)*cosh(x) (with 6 leading zeros).
a(n) = ceiling(binomial(n+6,6)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 89/5 - 24*log(2).
Sum_{n>=0} (-1)^n/a(n) = 5832*log(3/2) - 11819/5. (End)
EXAMPLE
a(0) = (2^(-1) + 0^0/2)*binomial(6,0) = 2*(1/2) = 1 (use 0^0 = 1).
MAPLE
[seq (ceil(binomial(n+6, 6)*2^(n-1)), n=0..22)]; # Zerinvary Lajos, Nov 01 2006
MATHEMATICA
Drop[With[{nmax = 56}, CoefficientList[Series[x^6*Exp[x]*Cosh[x]/6!, {x, 0, nmax}], x]*Range[0, nmax]!], 5] (* or *) Join[{1}, Table[2^(n-1)* Binomial[n+6, n], {n, 1, 30}] (* G. C. Greubel, Feb 05 2018 *)
LinearRecurrence[{14, -84, 280, -560, 672, -448, 128}, {1, 7, 56, 336, 1680, 7392, 29568, 109824}, 30] (* Harvey P. Dale, Jul 18 2023 *)
PROG
(PARI) my(x='x+O('x^30)); Vec(serlaplace(x^6*exp(x)*cosh(x)/6!)) \\ G. C. Greubel, Feb 05 2018
(Magma) [(2^(n-1) + 0^n/2)*Binomial(n+6, n): n in [0..30]]; // G. C. Greubel, Feb 05 2018
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 06 2003
STATUS
approved