A154986 → γ → θ+1

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Triangle begins:
  0: 1;
  1: 1;
  2: 1,   4;
  3: 1,  16;
  4: 1,  40,    72;
  5: 1,  80,   552;
  6: 1, 140,  2352,   2880;
  7: 1, 224,  7392,  33792;
  8: 1, 336, 19152, 209408,  201600;
  9: 1, 480, 43344, 919040, 3242880;

The row polynomials satisfy (n-2)*P(n,x) = -(n^5-8*n^4+23*n^3-28*n^2+12*n)*x^2*P(n-4,x) - (2*n^3-6*n^2+4*n)*x*P(n-3,x) - (n-(2*n^3-6*n^2+4*n)*x)*P(n-2,x) + (2*n-2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 4*x, P(3,x) = 1 + 16*x.

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[3, x_] := 1 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 8*n^4 + 23*n^3 - 28*n^2 + 12*n)*x^2*P[n - 4, x] - (2*n^3 - 6*n^2 + 4*n)*x*P[n - 3, x] - (n - (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 2, x] + (2*n - 2)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1 + 4*x; v[4] = 1 + 16*x; for(n = 4, N, my(i = n + 1); v[i] = (-(n^5 - 8*n^4 + 23*n^3 - 28*n^2 + 12*n)*x^2*v[i - 4] - (2*n^3 - 6*n^2 + 4*n)*x*v[i - 3] - (n - (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 2] + (2*n - 2)*v[i - 1])/(n - 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

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from gamma_polynomial of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), re…?
    • Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›

      • from Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.?
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.?
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • ↑ Liu–Wang theorem with a lag row (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.