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 A082140 A transform of binomial(n,6). 11
 1, 7, 56, 336, 1680, 7392, 29568, 109824, 384384, 1281280, 4100096, 12673024, 38019072, 111132672, 317521920, 889061376, 2444918784, 6615662592, 17641766912, 46425702400, 120706826240, 310388981760, 790081044480 (list; graph; refs; listen; history; text; internal format)
 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 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 *) PROG (PARI) 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 Cf. A082139, A080951. For n>0, a(n) = 1/2 * A002409(n). Sequence in context: A047664 A055345 A180287 * A264693 A054614 A270240 Adjacent sequences:  A082137 A082138 A082139 * A082141 A082142 A082143 KEYWORD easy,nonn AUTHOR Paul Barry, Apr 06 2003 STATUS approved

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Last modified June 25 07:53 EDT 2019. Contains 324347 sequences. (Running on oeis4.)