A075497 → reverse → θ+1

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Triangle begins:
  0: 1;
  1: 1,   4;
  2: 1,  12,   12;
  3: 1,  24,   84,     32;
  4: 1,  40,  300,    480,      80;
  5: 1,  60,  780,   2880,    2480,     192;
  6: 1,  84, 1680,  11200,   24080,   12096,      448;
  7: 1, 112, 3192,  33600,  136080,  185472,    56896,    1024;
  8: 1, 144, 5544,  84672,  556080, 1491840,  1355200,  261120,    2304;
  9: 1, 180, 9000, 188160, 1826160, 8164800, 15279040, 9553920, 1177344, 5120;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 1 + (4*n^2 - 4)*x + (4*n^3 - 4*n)*x^2)*v[i - 2] + ((2*n + 2)*x^2 + (4*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) + (2*n + 1 + 4*n^2*x)*v[i - 1] - 2*n*x^2*deriv(v[i - 1], x))/(n)); 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 reversal of Stirling2 triangle with scaled diagonals (powers of 2).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Stirling2 triangle with scaled diagonals (powers of 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 Stirling2 triangle with scaled diagonals (powers of 2).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Stirling2 triangle with scaled diagonals (powers of 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
        • ↑ Iterated real-rootedness preservers (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (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.