A204533
Triangle T(n,k), read by rows, given by (0, 1, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 8, 7, 3, 1, 0, 21, 22, 12, 4, 1, 0, 55, 67, 43, 18, 5, 1, 0, 144, 200, 147, 72, 25, 6, 1, 0, 377, 588, 486, 271, 110, 33, 7, 1, 0, 987, 1708, 1566, 976, 450, 158, 42, 8, 1
Offset: 0
Triangle begins:
1;
0, 1;
0, 1, 1;
0, 3, 2, 1;
0, 8, 7, 3, 1;
0, 21, 22, 12, 4, 1;
0, 55, 67, 43, 18, 5, 1;
0, 144, 200, 147, 72, 25, 6, 1;
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Table[Sum[Binomial[k, m - 1] Binomial[n - 2 m + k, n - k - 1], {k, 0, n - 1}] + Boole[n == m == 0], {n, 0, 9}, {m, 0, n}] // Flatten (* Michael De Vlieger, Sep 26 2018 *)
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T(n,m):= if n=0 and m=0 then 1 else sum(binomial(k,m-1)*binomial(n-2*m+k,n-k-1),k,0,n-1); /* Vladimir Kruchinin, Sep 27 2018 */
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T(n,k) = if ((n==0) && (k==0), 1, sum(i=0, n-1, binomial(i,k-1)*binomial(n-2*k+i,n-i-1))); \\ Michel Marcus, Sep 27 2018
A111053
Number of permutations which avoid the patterns 1324 and (2143 with Bruhat restriction {2<->3}). Also the number of permutations whose graphs are acyclic.
Original entry on oeis.org
1, 2, 6, 22, 89, 379, 1661, 7405, 33367, 151398, 690147, 3156112, 14465746, 66409493, 305232025, 1404129530, 6463476538, 29767212095, 137142651679, 632021380433, 2913316615372, 13431328632593, 61931182541194, 285592218851606, 1317104663887309, 6074682489939359, 28018852961838675, 129239701278757210
Offset: 1
x + 2*x^2 + 6*x^3 + 22*x^4 + 89*x^5 + 379*x^6 + 1661*x^7 + 7405*x^8 + ...
- S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. See p. 399, Table A.7.
- H. Abe and S. Billey, Consequences of the Lakshmibai-Sandhya theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry, 2014.
- M. Bousquet-Mélou and S. Butler, Forest-like permutations, arXiv:math/0603617 [math.CO], 2006.
- S. Butler, On permutations which are 1324 and {overline 2143} avoiding, 2005.
- S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).
- Haruhisa Enomoto, Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras, arXiv:2002.09205 [math.RT], 2020.
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a = DifferenceRoot[Function[{a, n}, {(4n^2 + 46n + 60)a[n] + (-9n^2 - 105n - 156)a[n+1] + (22n^2 + 256n + 372)a[n+2] + (-9n^2 - 111n - 240)a[n+3] + (n+4)(n+9)a[n+4] == 0, a[1] == 1, a[2] == 2, a[3] == 6, a[4] == 22}]];
Array[a, 28] (* Jean-François Alcover, Dec 17 2018 *)
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x='x+O('x^66);
gf=((1-x)*(1-4*x-2*x^2)-(1-5*x)*sqrt(1-4*x))/(2*(1-5*x+2*x^2-x^3));
Vec(gf) /* Joerg Arndt, Jun 25 2011 */
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{a(n) = if( n<0, 0, polcoeff( 2 * x * (1 - 4*x - x^2) / ((1 - x) * (1 - 4*x - 2*x^2) + (1 - 5*x) * sqrt(1 - 4*x + x * O(x^n))), n))} /* Michael Somos, Jan 12 2012 */
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