A081200 6th binomial transform of (0,1,0,1,0,1,...), A000035.

0, 1, 12, 109, 888, 6841, 51012, 372709, 2687088, 19200241, 136354812, 964249309, 6798573288, 47834153641, 336059778612, 2358521965909, 16540171339488, 115933787267041, 812299450322412, 5689910849522509, 39848449432985688, 279034513462540441, 1953718431395986212

Offset: 0

Author

Paul Barry, Mar 11 2003

Keywords

• nonn
• easy
• changed

Comments

Binomial transform of A081199.

Conjecture (verified up to a(9)): Number of collinear 5-tuples of points in a 5 X 5 X 5 X ... n-dimensional cubic grid. - Ron Hardin, May 24 2010

a(n) is also the total number of words of length n, over an alphabet of seven letters, of which one of them appears an odd number of times. See the Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter case), and the Balakrishnan reference there. For the 2-, 3-, 5-, 6- and 8-letter case analogs see A131577, A003462, A005059, A081199, A081201 respectively. - Wolfdieter Lang, Jul 17 2017

Examples

			The a(2) = 12 words of length 2 over {A, B, C, D, E, F, G} with say, A, appearing an odd number of times (that is once) are: AB, AC, AD, AE, AF, AG; BA, CA, DA, EA, FA, GA. - _Wolfdieter Lang_, Jul 17 2017
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
• Index entries for linear recurrences with constant coefficients, signature (12,-35).

Crossrefs

Cf. A000035, A003462, A005059, A006516, A081199, A081201 (binomial transform, and 8-letter analog), A121213, A131577.

Apart from offset same as A016161.

Programs

Magma

[7^n/2-5^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

Mathematica

CoefficientList[Series[x / ((1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{12,-35},{0,1},30] (* Harvey P. Dale, Feb 07 2014 *)

Sage

[lucas_number1(n,12,35) for n in range(0, 21)] # Zerinvary Lajos, Apr 27 2009

Formula

a(n) = 12*a(n-1) - 35*a(n-2), a(0) = 0, a(1) = 1.

G.f.: x/((1-5*x)*(1-7*x)).

a(n) = 7^n/2 - 5^n/2.

a(n) = Sum_{k=0..n-1} 7^k * 5^(n-k-1), with a(0)=0. - Reinhard Zumkeller, Aug 01 2010

a(n) = A121213(n)/2. - Reinhard Zumkeller, Aug 01 2010

E.g.f.: exp(5*x)*(exp(2*x) - 1)/2. - Stefano Spezia, Jun 19 2021