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A081200
6th binomial transform of (0,1,0,1,0,1,...), A000035.
12
0, 1, 12, 109, 888, 6841, 51012, 372709, 2687088, 19200241, 136354812, 964249309, 6798573288, 47834153641, 336059778612, 2358521965909, 16540171339488, 115933787267041, 812299450322412, 5689910849522509, 39848449432985688
OFFSET
0,3
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
LINKS
Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
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
EXAMPLE
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
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 *)
PROG
(Sage) [lucas_number1(n, 12, 35) for n in range(0, 21)] # Zerinvary Lajos, Apr 27 2009
(Magma) [7^n/2-5^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013
CROSSREFS
Cf. A000035, A003462, A005059, A006516, A016161, A081199, A081201 (binomial transform, and 8-letter analog), A121213, A131577.
Sequence in context: A128877 A085797 A016161 * A351161 A016214 A037581
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Mar 11 2003
STATUS
approved