

A032093


Number of reversible strings with n1 beads of 2 colors. 6 beads are black. Strings are not palindromic.


5



3, 12, 40, 100, 226, 452, 848, 1484, 2485, 3976, 6160, 9240, 13524, 19320, 27072, 37224, 50391, 67188, 88440, 114972, 147862, 188188, 237328, 296660, 367913, 452816, 553504, 672112, 811240, 973488, 1161984, 1379856
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OFFSET

8,1


COMMENTS

From Petros Hadjicostas, May 19 2018: (Start)
Let k be an integer >= 2. The g.f. of the BHK[k] transform of the sequence (c(n): n>=1), with g.f. C(x) = Sum_{n>=1} c(n)*x^n, is A_k(x) = (C(x)^k  C(x^2)^(k/2))/2 if k is even, and A_k(x) = (C(x)/2)*(C(x)^{k1}  C(x^2)^{(k1)/2}) if k is odd. This follows easily from the formulae in C. G. Bower's web link below about transforms.
When k is odd and c(n) = 1 for all n>=1, we get C(x) = x/(1x) and A_k(x) = (1/2)*(x/(1x))*((x/(1x))^{k1}  (x^2/(1x^2))^{(k1)/2}). If (a_k(n): n>=1) is the output sequence (with g.f. A_k(x)), then it can be proved (using Taylor expansions) that a_k(n) = (1/2)*(binomial(n1, nk)  binomial(floor((n1)/2), floor((nk)/2))) for n >= k+1. (Clearly, a_k(1) = ... = a_k(k) = 0.)
In this sequence, k = 7, and (according to C. G. Bower) a(n) = a_{k=7}(n) is the number of reversible nonpalindromic compositions of n with 7 positive parts. If n = b_1 + b_2 + b_3 + b_4 + b_5 + b_6 + b_7 is such a composition of n (with b_i >=1), then it is equivalent to the composition n = b_7 + b_6 + b_5 + b_4 + b_3 + b_2 + b_1, and each equivalent class has two elements because here linear palindromes are not allowed as compositions of n.
The fact that we are finding the BHK[7] transform of 1, 1, 1, ... means that each part of each composition of n can have exactly one color (see Bower's link below about transforms).
In each such composition replace each b_i with one black (B) ball followed by b_i  1 white (W) balls. Then drop the first black (B) ball. We then get a reversible nonpalindromic string of length n1 that has 6 black balls and n7 white balls. This process, applied to the equivalent compositions n = b_1 + b_2 + b_3 + b_4 + b_5 + b_6 + b_7 = b_7 + b_6 + b_5 + b_4 + b_3 + b_2 + b_1, gives two strings of length n1 with 6 black balls and n7 white balls that are mirror images of each other.
Hence, for n>=2, a(n) = a_{k=7}(n) is also the number of reversible nonpalindromic strings of length n1 that have k1 = 6 black balls and nk = n7 white balls. (Clearly, a(n) = a_{k=7}(n) > 0 only for n >= 8. For n=7, the composition 1+1+1+1+1+1+1, which corresponds to string BBBBBB, is discarded because it is palindromic.)
(End)


LINKS

Table of n, a(n) for n=8..39.
C. G. Bower, Transforms (2)


FORMULA

"BHK[ 7 ]" (reversible, identity, unlabeled, 7 parts) transform of 1, 1, 1, 1, ...
Empirical G.f.: x^8*(x^2+3)/((x1)^7*(x+1)^3).  Colin Barker, Nov 24 2012
From Petros Hadjicostas, May 19 2018: (Start)
a(n) = (1/2)*(binomial(n1, n7)  binomial(floor((n1)/2), floor((n7)/2))) for n >= 8.
G.f.: (1/2)*(x/(1x))*((x/(1x))^6  (x^2/(1x^2))^3), which is the same as the g.f. given by Colin Barker above.
(End)


EXAMPLE

From Petros Hadjicostas, May 19 2018: (Start)
For n=8, we have the following 3 reversible nonpalindromic compositions with 7 parts of n: 1+1+1+1+1+1+2 (= 2+1+1+1+1+1+1), 1+1+1+1+1+2+1 (= 1+2+1+1+1+1+1), and 1+1+1+1+2+1+1 (= 1+1+2+1+1+1+1). Using the process described in the comments, we get the following reversible nonpalindromic strings with 6 black balls and n7=1 white balls: BBBBBBW (= WBBBBBB), BBBBBWB (= BWBBBBB), and BBBBWBB (= BBWBBBB).
For n=9, we get the following 12 compositions and 12 corresponding strings:
1+1+1+1+1+1+3 <> BBBBBBWW
1+1+1+1+1+3+1 <> BBBBBWWB
1+1+1+1+3+1+1 <> BBBBWWBB
1+1+1+1+1+2+2 <> BBBBBWBW
1+1+1+1+2+1+2 <> BBBBWBBW
1+1+1+2+1+1+2 <> BBBWBBBW
1+1+2+1+1+1+2 <> BBWBBBBW
1+2+1+1+1+1+2 <> BWBBBBBW
1+1+1+1+2+2+1 <> BBBBWBWB
1+1+1+2+1+2+1 <> BBBWBBWB
1+1+2+1+1+2+1 <> BBWBBBWB
1+1+1+2+2+1+1 <> BBBWBWBB
(End)


CROSSREFS

Cf. A002620, A006584, A032091, A032092, A032094, A239572, A282011.
Sequence in context: A303348 A237036 A034956 * A007993 A293366 A080929
Adjacent sequences: A032090 A032091 A032092 * A032094 A032095 A032096


KEYWORD

nonn


AUTHOR

Christian G. Bower


EXTENSIONS

Definition changed slightly by Harvey P. Dale, Oct 02 2017


STATUS

approved



