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 A217205 Triangle read by rows, related to Bell numbers A000110. 0
 1, 1, 1, 1, 2, 1, 4, 7, 5, 2, 11, 28, 28, 16, 5, 41, 131, 153, 118, 71, 16, 162, 690, 872, 892, 759, 272, 61, 715, 4033, 5191, 7060, 7262, 3468, 1665, 272, 3425, 25864, 32398, 58608, 66510, 41088, 29778, 7936, 1385, 17722, 180265, 211937, 510812, 601080, 479772, 443231, 156176, 60991, 7936 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS See Ma and Chow (2012) for precise definition (cf. On combinations of polynomials and Euler numbers). LINKS Table of n, a(n) for n=1..55. Shi-Mei Ma and Chak-On Chow, Enumeration of permutations by number of cyclic peaks and cyclic valleys, arXiv preprint arXiv:1203.6264 [math.CO], 2012. EXAMPLE Triangle begins: 1 1,1 1,2,1 4,7,5,2 11,28,28,16,5 41,131,153,118,71,16 162,690,872,892,759,272,61 ... MATHEMATICA P[1] := x y; P[n_] := P[n] = ((n-1) q + x y) P[n-1] + 2 q (1-q) D[P[n-1], q] + x (1-q) D[P[n-1], x] + (1-y) D[P[n-1], y] // Simplify; V[1] = x y; V[n_] := V[n] = ((n-1) q + x y) V[n-1] + 2 q (1-q) D[V[n-1], q] + 2 x (1-q) D[V[n-1], x] + (1 - 2 y + q y) D[V[n-1], y] // Simplify; Dn[n_] := P[n] /. {x -> 1, y -> 0}; Dbar[n_] := V[n] /. {x -> 1, y -> 0}; Inq[1] = 1; Inq[n_] := (Dn[n] /. q -> q^2) + q (Dbar[n] /. q -> q^2); Table[CoefficientList[Inq[n], q], {n, 1, 10}] // Flatten (* Jean-François Alcover, Sep 25 2018 *) PROG (PARI) tabl(m) = { J = 1; for (d=0, poldegree(J, q), print1(polcoeff(J, d, q), ", "); ); print(""); Pa = x; Pb = x; Pa1 = subst(Pa, x, 1); Pb1 = subst(Pb, x, 1); J = subst(Pa1, q, q^2) + q*subst(Pb1, q, q^2); for (d=0, poldegree(J, q), print1(polcoeff(J, d, q), ", "); ); print(""); Qa = (1+q)*x; Qb = 2*x; for (n=3, m, Qa1 = subst(Qa, x, 1); Qb1 = subst(Qb, x, 1); J = subst(Qa1, q, q^2) + q*subst(Qb1, q, q^2); for (d=0, poldegree(J, q), print1(polcoeff(J, d, q), ", "); ); print(""); newPa = n*q*Qa + 2*q*(1-q)*deriv(Qa, q) + x*(1-q)*deriv(Qa, x) + n*x*Pa; newPb = n*q*Qb + 2*q*(1-q)*deriv(Qb, q) + 2*x*(1-q)*deriv(Qb, x) + n*x*Pb; Pa = Qa; Qa = newPa; Pb = Qb; Qb = newPb; ); } \\ Michel Marcus, Feb 11 2013 CROSSREFS First column appears to be A032265. Sequence in context: A072015 A123242 A322941 * A358566 A139769 A357470 Adjacent sequences: A217202 A217203 A217204 * A217206 A217207 A217208 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, Sep 27 2012 EXTENSIONS More terms from Michel Marcus, Feb 11 2013 STATUS approved

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Last modified April 21 06:29 EDT 2024. Contains 371850 sequences. (Running on oeis4.)