A159854 → x ↦ x+1 → θ+1

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Triangle begins:
  0:   1,    4,     3;
  1:   2,   10,    12,     4;
  2:   4,   24,    39,    24,     5;
  3:   8,   56,   114,   100,    40,     6;
  4:  16,  128,   312,   352,   205,    60,     7;
  5:  32,  288,   816,  1120,   850,   366,    84,     8;
  6:  64,  640,  2064,  3328,  3100,  1752,   595,   112,    9;
  7: 128, 1408,  5088,  9408, 10360,  7224,  3234,   904,  144,   10;
  8: 256, 3072, 12288, 25600, 32480, 26880, 14896,  5504, 1305,  180,  11;
  9: 512, 6656, 29184, 67584, 96960, 92736, 61152, 28032, 8802, 1810, 220, 12;

The row polynomials satisfy (2+(n+6)*x+(n+2)*x^2)*P(n,x) = (4+(2*n+16)*x+(3*n+13)*x^2+(n+3)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 4*x + 3*x^2.

P[0, x_] := 1 + 4*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4 + (2*n + 16)*x + (3*n + 13)*x^2 + (n + 3)*x^3)*P[n - 1, x])/(2 + (n + 6)*x + (n + 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 4*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4 + (2*n + 16)*x + (3*n + 13)*x^2 + (n + 3)*x^3)*v[i - 1])/(2 + (n + 6)*x + (n + 2)*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 shift_x_plus_one of Riordan array (1-x,x/(1-x)).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Riordan array (1-x,x/(1-x)).?
        • 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
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    • Geometric basis map (elementary) polytool ? step ›

      • from Riordan array (1-x,x/(1-x)).?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.