A227450 → reverse → θ+1

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Triangle begins:
  0:    2;
  1:    4,     4;
  2:    8,    24,      6;
  3:   16,    96,     84,       8;
  4:   32,   320,    600,     240,      10;
  5:   64,   960,   3120,    2880,     620,      12;
  6:  128,  2688,  13440,   22400,   12040,    1512,      14;
  7:  256,  7168,  51072,  134400,  136080,   46368,    3556,     16;
  8:  512, 18432, 177408,  677376, 1112160,  745920,  169400,   8160,    18;
  9: 1024, 46080, 576000, 3010560, 7304640, 8164800, 3819760, 597120, 18396, 20;

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

P[0, x_] := 2;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n + 4 + (4*n^2 - 4)*x + (n^3 - n)*x^2)*P[n - 2, x] + ((2*n + 2)*x^2 + (n^2 + n)*x^3)*D[P[n - 2, x], x] + (4*n + 2 + 2*n^2*x)*P[n - 1, x] - 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] = 2; v[2] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n + 4 + (4*n^2 - 4)*x + (n^3 - n)*x^2)*v[i - 2] + ((2*n + 2)*x^2 + (n^2 + n)*x^3)*deriv(v[i - 2], x) + (4*n + 2 + 2*n^2*x)*v[i - 1] - 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 Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.?
                • ↑ 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.