A391533 → basis A013609 → θ+1

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Submission text ?

Triangle begins:
  0:    9,    24,     12;
  1:   18,    84,     96,      32;
  2:   36,   240,    444,     320,      80;
  3:   72,   624,   1608,    1824,     960,     192;
  4:  144,  1536,   5088,    7936,    6480,    2688,     448;
  5:  288,  3648,  14784,   29440,   32800,   20928,    7168,    1024;
  6:  576,  8448,  40512,   98304,  139200,  120576,   63168,   18432,    2304;
  7: 1152, 19200, 106368,  304640,  524160,  575232,  407680,  181248,   46080,    5120;
  8: 2304, 43008, 270336,  892928, 1809920, 2408448, 2157568, 1294336,  499968,  112640,  11264;
  9: 4608, 95232, 669696, 2506752, 5852160, 9160704, 9934848, 7520256, 3912192, 1336320, 270336, 24576;

The row polynomials satisfy (3+(3*n+6)*x+(2*n+4)*x^2)*P(n,x) = (6+(6*n+24)*x+(10*n+30)*x^2+(4*n+12)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 24*x + 12*x^2.

P[0, x_] := 9 + 24*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6 + (6*n + 24)*x + (10*n + 30)*x^2 + (4*n + 12)*x^3)*P[n - 1, x])/(3 + (3*n + 6)*x + (2*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 9 + 24*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6 + (6*n + 24)*x + (10*n + 30)*x^2 + (4*n + 12)*x^3)*v[i - 1])/(3 + (3*n + 6)*x + (2*n + 4)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
  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_A013609 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    • Geometric basis map (elementary) polytool ? step ›

      • from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
                • ↑ Liu–Wang theorem (above)
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
                • Parent-row provers of polytool polytool ? step › direct check
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (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.