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A354786
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Irregular triangle read by rows: T(n,k) is the number of anti-palindromic compositions of n of length k, n >= 0, 0 <= k <= floor((2*n+1)/3).
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2
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1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 2, 0, 1, 4, 4, 0, 1, 4, 8, 4, 0, 1, 6, 12, 8, 4, 0, 1, 6, 18, 20, 12, 0, 1, 8, 24, 32, 32, 8, 0, 1, 8, 32, 56, 64, 24, 8, 0, 1, 10, 40, 80, 120, 72, 32, 0, 1, 10, 50, 120, 200, 152, 104, 16, 0, 1, 12, 60, 160, 320, 312, 256, 64, 16
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OFFSET
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0,8
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COMMENTS
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A composition S with sum n and length k is anti-palindromic if S(i) != S(k+1-i) for 1 <= i < floor(k). - Andrew Howroyd, Feb 28 2023
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LINKS
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George E. Andrews, Matthew Just, and Greg Simay, Anti-palindromic compositions, arXiv:2102.01613 [math.CO], 2021. Also Fib. Q., 60:2 (2022), 164-176. See Table 2.
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FORMULA
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G.f.: A(x,y) = (1 + x*y/(1 - x))/(1 - 2*x^3*y^2/((1 + x)*(1 - x)^2)). - Andrew Howroyd, Feb 28 2023
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EXAMPLE
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Triangle begins:
1;
0, 1;
0, 1,
0, 1, 2;
0, 1, 2, 2;
0, 1, 4, 4;
0, 1, 4, 8, 4;
0, 1, 6, 12, 8, 4;
0, 1, 6, 18, 20, 12;
...
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PROG
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(PARI) T(n)=[Vecrev(p) | p<-Vec((1 + x*y/(1 - x))/(1 - 2*x^3*y^2/((1 + x)*(1 - x)^2)) + O(x*x^n))]
{ my(rows=T(12)); for(i=1, #rows, print(rows[i])) } \\ Andrew Howroyd, Feb 28 2023
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CROSSREFS
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Row sums are Tribonacci numbers (A000213).
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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