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 A162206 Triangle read by rows in which row n (n >= 1) gives coefficients in expansion of the polynomial f(n) * (Product_{i=1..n-1} f(2i))/ f(1)^n, where f(k) = 1 - x^k. 49
 1, 1, 2, 1, 1, 3, 5, 6, 5, 3, 1, 1, 4, 9, 16, 23, 28, 30, 28, 23, 16, 9, 4, 1, 1, 5, 14, 30, 54, 85, 120, 155, 185, 205, 212, 205, 185, 155, 120, 85, 54, 30, 14, 5, 1, 1, 6, 20, 50, 104, 190, 314, 478, 679, 908, 1151, 1390, 1605, 1776, 1886, 1924, 1886, 1776 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS For n >= 3, this polynomial is the Poincaré polynomial (or growth series) for the reflection group (or Weyl group, or finite Coxeter group) D_n. Row lengths are 1, 3, 7, 13, 21, 31, 43, 57, ...: see A002061. - Michel Marcus, May 17 2013 The asymptotic growth of maximum elements for the reflection group D_n is about 2(n-1/2) (compare with A000140). - Mikhail Gaichenkov, Aug 21 2019 REFERENCES N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10a, page 231, W(t). J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59. LINKS Jean-François Alcover, Table of n, a(n) for n = 1..9020 [30 rows] Marwa Ben Abdelmaksoud, Adel Hamdi, width-k Eulerian polynomials of type A and B and its Gamma-positivity, arXiv:1912.08551 [math.CO], 2019. M. Gaichenkov, The growth of maximum elements for the reflection group \$D_n\$, MathOverflow, 2019. M. Rubey, St001443: Finite Cartan types ⟶ ℤ, StatisticsDatabase, 2019. EXAMPLE Triangle begins: 1 1,2,1, 1,3,5,6,5,3,1, 1,4,9,16,23,28,30,28,23,16,9,4,1, 1,5,14,30,54,85,120,155,185,205,212,205,185,155,120,85,54,30,14,5,1, 1,6,20,50,104,190,314,478,679,908,1151,1390,1605,1776,1886,1924,1886,1776,1605,1390,1151,908,679,478,314,190,104,50,20,6,1, 1,7,27,77,181,371,686,1169,1862,2800,4005,5481,7210,9149,11230,13363,15442,17353,18983,20230,21013,21280,21013,20230,18983,17353,15442,13363,11230,9149,7210,5481,4005,2800,1862,1169,686,371,181,77,27,7,1, MATHEMATICA T[nn_] := Reap[Do[x = y + y O[y]^(n^2); v = (1 - x^n) Product[1 - x^(2k), {k, 1, n - 1}]/(1 - x)^n // CoefficientList[#, y]&; Sow[v], {n, nn}]][[2, 1]]; T[6] // Flatten (* Jean-François Alcover, Mar 25 2020, after PARI *) PROG (PARI) tabl(nn) = {for (n=1, nn, x = y+y*O(y^(n^2)); v = Vec((1-x^n)*prod(k=1, n-1, 1-x^(2*k))/(1-x)^n); for (i=1, #v, if (v[i], print1(v[i], ", ")); ); print(); ); } \\ Michel Marcus, May 17 2013 CROSSREFS The growth series for D_k, k >= 5, that is, rows 5 through 12 of this triangle, are A162208-A162212, A162248, A162288, A162297. Cf. A002061. Sequence in context: A210098 A241188 A145236 * A075248 A336707 A128325 Adjacent sequences:  A162203 A162204 A162205 * A162207 A162208 A162209 KEYWORD nonn,tabf AUTHOR John Cannon and N. J. A. Sloane, Nov 30 2009 EXTENSIONS Revised by N. J. A. Sloane, Jan 10 2016 STATUS approved

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Last modified August 7 22:44 EDT 2020. Contains 336279 sequences. (Running on oeis4.)