A082140 A transform of binomial(n,6).

1, 7, 56, 336, 1680, 7392, 29568, 109824, 384384, 1281280, 4100096, 12673024, 38019072, 111132672, 317521920, 889061376, 2444918784, 6615662592, 17641766912, 46425702400, 120706826240, 310388981760, 790081044480

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

From Amiram Eldar, Jan 07 2022: (Start)

Sum_{n>=0} 1/a(n) = 89/5 - 24*log(2).

Sum_{n>=0} (-1)^n/a(n) = 5832*log(3/2) - 11819/5. (End)

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 *)
LinearRecurrence[{14, -84, 280, -560, 672, -448, 128}, {1, 7, 56, 336, 1680, 7392, 29568, 109824}, 30] (* Harvey P. Dale, Jul 18 2023 *)

Prog

(PARI) my(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. A082137, 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