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%I #27 Sep 08 2022 08:45:09
%S 1,8,72,480,2640,12672,54912,219648,823680,2928640,9957376,32587776,
%T 103194624,317521920,952565760,2794192896,8033304576,22682271744,
%U 63006310400,172438323200,465583472640,1241555927040,3273192898560
%N A transform of C(n,7).
%C 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.
%H G. C. Greubel, <a href="/A082141/b082141.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (16,-112,448,-1120,1792,-1792, 1024,-256).
%F a(n) = (2^(n-1) + 0^n/2)*C(n+7,n).
%F a(n) = Sum_{j=0..n} C(n+7, j+7)*C(j+7, 7)*(1+(-1)^j)/2.
%F 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.
%F E.g.f.: (x^7/7!)*exp(x)*cosh(x) (with 7 leading zeros).
%F a(n) = ceiling(binomial(n+7,7)*2^(n-1)). - _Zerinvary Lajos_, Nov 01 2006
%F From _Amiram Eldar_, Jan 07 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 28*log(2) - 274/15.
%F Sum_{n>=0} (-1)^n/a(n) = 20412*log(3/2) - 124132/15. (End)
%e a(0) = (2^(-1) + 0^0/2)*C(7,0) = 2*(1/2) = 1 (using 0^0=1).
%p [seq (ceil(binomial(n+7,7)*2^(n-1)),n=0..22)]; # _Zerinvary Lajos_, Nov 01 2006
%t 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 *)
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(x^7*exp(x)*cosh(x)/7!)) \\ _G. C. Greubel_, Feb 05 2018
%o (Magma) [(2^(n-1) + 0^n/2)*Binomial(n+7,n): n in [0..30]]; // _G. C. Greubel_, Feb 05 2018
%Y Cf. A054851, A082137, A082139, A082140.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Apr 06 2003
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