A370767
Number of signed permutations of length n+1 with adjacent elements differing by more than 1 and whose first element is 1.
Original entry on oeis.org
1, 1, 3, 17, 139, 1401, 16867, 236513, 3787707, 68219081, 1364931859, 30037136433, 721044433387, 18750182814233, 525071095004739, 15753703863875201, 504159100060894747, 17142539126080474473, 617165134818228049267, 23453349764127439545041
Offset: 0
In the following examples, the number of assignments of signs to each unsigned permutation is shown in parenthesis.
a(2) = 3: 123(1), 132(2). Total is 1 + 2 = 3.
a(3) = 17: 1234(1), 1243(2), 1324(4), 1342(4), 1423(4), 1432(2).
A370768
Number of signed permutations of length n+2 with adjacent elements differing by more than 1 whose first element is 1 and whose last element has absolute value n+2.
Original entry on oeis.org
1, 1, 5, 29, 249, 2553, 31181, 441845, 7133569, 129304593, 2600559125, 57473713741, 1384615153033, 36115750475433, 1014026439534045, 30493381288216357, 977824818833573137, 33307253433327375809, 1201023016203128722725, 45705676512051750367357
Offset: 0
In the following examples, the number of assignments of signs to each unsigned permutation is shown in parenthesis.
a(0) = 1 from the signed permutation (1, -2).
a(1) = 1 from the signed permutation (1, -2, 3).
a(2) = 5: 1234(1), 1324(4). Total is 1 + 4 = 5.
a(3) = 29: 12345(1), 12435(4), 13245(4), 13425(8), 14235(8), 14325(4).
The 2*a(3) = 58 signed permutations of length 4 with adjacent elements differing by more than 1 which start with +-1 or end with +-4 are: 1234(2), 1243(4), 1324(8), 1342(8), 1423(8), 1432(4), 2134(4), 2314(8), 3124(8), 3214(4).
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a(n)=subst(serlaplace(polcoef(2/((1 + x)*(1 + (1 - 2*y)*x + 2*y*x^2)) - 1/(1 + x) + O(x*x^n), n)), y, 1)
A370769
Number of achiral unicursal star polygons (no edge joins adjacent vertices) that can be formed by connecting the vertices of a regular n-gon.
Original entry on oeis.org
0, 0, 0, 0, 1, 1, 5, 11, 49, 123, 521, 1583, 6581, 23239, 95509, 384771, 1570265, 7106995, 28869825, 145034327, 587270877, 3242792607, 13100475021, 78866628011, 318067071169, 2073381189259, 8350998470777, 58602568320255, 235794888434053, 1772311322357623
Offset: 1
Cf.
A231091 (stars up to rotation),
A370459 (up to rotation and reflection).
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Ro(n)=-(-1)^n + subst(serlaplace(polcoef(((1 - x)^2)/(2*(1 + x)*(1 + (1 - 2*y)*x + 2*y*x^2)) + O(x*x^n), n)), y, 1)
Re(n)=subst(serlaplace(polcoef((1 - 3*x)/(8*(1 + (1 - 2*y)*x + 2*y*x^2)) + O(x*x^n), n)), y, 1)
a(n) = if(n < 3, 0, if(n % 2, Ro(n\2), Re(n/2)))
Showing 1-3 of 3 results.
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