A082141 A transform of C(n,7).

1, 8, 72, 480, 2640, 12672, 54912, 219648, 823680, 2928640, 9957376, 32587776, 103194624, 317521920, 952565760, 2794192896, 8033304576, 22682271744, 63006310400, 172438323200, 465583472640, 1241555927040, 3273192898560

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

From Amiram Eldar, Jan 07 2022: (Start)

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)

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) 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

Crossrefs

Cf. A054851, A082137, A082139, A082140.

Sequence in context: A189954 A271028 A180288 * A304826 A270241 A054615

Adjacent sequences: A082138 A082139 A082140 * A082142 A082143 A082144

Keyword

easy,nonn

Author

Paul Barry, Apr 06 2003

Status

approved