A161198 → basis A013610 → θ+1

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Submission text ?

Triangle begins:
   1:         1;
   2:         3,         12;
   3:        15,         96,         108;
   4:       105,        852,        1620,         864;
   5:       945,       8928,       22248,       20736,        6480;
   6:     10395,     109548,      325080,      406080,      226800,       46656;
   7:    135135,    1548864,     5211972,     7879680,     5994000,     2239488,      326592;
   8:   2027025,   24854580,    92119356,   159890976,   149007600,    76749120,    20575296,    2239488;
   9:  34459425,  446852160,  1789720272,  3455101440,  3732311520,  2377589760,   887023872,  179159040,   15116544;
  10: 654729075, 8903704140, 38026354608, 79964689536, 96827179680, 72046848384, 33496581888, 9486471168, 1496537856, 100776960;

The row polynomials satisfy P(n,x) = -(4*n^2-12*n+9+(24*n-36)*x+36*x^2)*P(n-2,x) + (4*n-4+12*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 12*x.

P[1, x_] := 1;
P[2, x_] := 3 + 12*x;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 9 + (24*n - 36)*x + 36*x^2)*P[n - 2, x] + (4*n - 4 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 12*x; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 12*n + 9 + (24*n - 36)*x + 36*x^2)*v[i - 2] + (4*n - 4 + 12*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A013610 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.