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A082141
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A transform of C(n,7).
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10
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1, 8, 72, 480, 2640, 12672, 54912, 219648, 823680, 2928640, 9957376, 32587776, 103194624, 317521920, 952565760, 2794192896, 8033304576, 22682271744, 63006310400, 172438323200, 465583472640, 1241555927040, 3273192898560
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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).
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)
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EXAMPLE
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a(0) = (2^(-1) + 0^0/2)*C(7,0) = 2*(1/2) = 1 (using 0^0=1).
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MAPLE
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[seq (ceil(binomial(n+7, 7)*2^(n-1)), n=0..22)]; # Zerinvary Lajos, Nov 01 2006
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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