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 A082141 A transform of C(n,7). 10
 1, 8, 72, 480, 2640, 12672, 54912, 219648, 823680, 2928640, 9957376, 32587776, 103194624, 317521920, 952565760, 2794192896, 8033304576, 22682271744, 63006310400, 172438323200, 465583472640, 1241555927040, 3273192898560 (list; graph; refs; listen; history; text; internal format)
 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 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) 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 Cf. A082140, A082139, A054851. Sequence in context: A189954 A271028 A180288 * A304826 A270241 A054615 Adjacent sequences:  A082138 A082139 A082140 * A082142 A082143 A082144 KEYWORD easy,nonn,changed AUTHOR Paul Barry, Apr 06 2003 STATUS approved

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Last modified June 25 10:10 EDT 2019. Contains 324351 sequences. (Running on oeis4.)