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A082141 A transform of C(n,7). 10

%I

%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

%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) 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. A082140, A082139, A054851.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Apr 06 2003

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Last modified July 23 03:00 EDT 2019. Contains 325230 sequences. (Running on oeis4.)