A161198 → basis A007318 → D+1

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Submission text ?

Triangle begins:
   1:          1;
   2:          5,          2;
   3:         31,         24,          4;
   4:        247,        262,         84,          8;
   5:       2433,       3136,       1400,        256,        16;
   6:      28653,      42338,      23320,       6000,       720,       32;
   7:     393279,     644216,     411916,     132160,     22480,     1920,       64;
   8:    6169455,   10966086,    7853244,    2952152,    631120,    76832,     4928,    128;
   9:  108934785,  207047232,  162260976,   68854016,  17369184,  2673664,   245504,  12288,   256;
  10: 2138679765, 4300717698, 3629624880, 1696731616, 486153696, 88799424, 10359552, 744960, 29952, 512;

The row polynomials satisfy P(n,x) = -(4*n^2-12*n+9+(8*n-12)*x+4*x^2)*P(n-2,x) + (4*n-4+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 2*x.

P[1, x_] := 1;
P[2, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 9 + (8*n - 12)*x + 4*x^2)*P[n - 2, x] + (4*n - 4 + 4*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] = 5 + 2*x; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 12*n + 9 + (8*n - 12)*x + 4*x^2)*v[i - 2] + (4*n - 4 + 4*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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A007318 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)
    • Affine, reversal and Möbius substitutions (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.