A122542 → reverse → θ+1

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Triangle begins:
   1: 1;
   2: 1;
   3: 1,  4;
   4: 1,  8,   6;
   5: 1, 12,  24,    8;
   6: 1, 16,  54,   48,   10;
   7: 1, 20,  96,  152,   80,   12;
   8: 1, 24, 150,  352,  330,  120,   14;
   9: 1, 28, 216,  680,  960,  612,  168,  16;
  10: 1, 32, 294, 1168, 2250, 2160, 1022, 224, 18;

The row polynomials satisfy -(n^2-n+(2*n^3-7*n^2+9*n)*x-(2*n^3-7*n^2+5*n-6)*x^2-(n^2-5*n+6)*x^3)*P(n,x) = -(n^2-n+(2*n^3+n^2-7*n)*x+(6*n^3-11*n^2+5*n-6)*x^2+(n^2-5*n+6)*x^3)*P(n-1,x) + (n*x+(2*n^2+6*n+1)*x^2+(12*n^2+2*n+5)*x^3+(2*n^2+6*n-5)*x^4+(n-1)*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (2*n^3 + n^2 - 7*n)*x + (6*n^3 - 11*n^2 + 5*n - 6)*x^2 + (n^2 - 5*n + 6)*x^3)*P[n - 1, x] + (n*x + (2*n^2 + 6*n + 1)*x^2 + (12*n^2 + 2*n + 5)*x^3 + (2*n^2 + 6*n - 5)*x^4 + (n - 1)*x^5)*D[P[n - 1, x], x])/(-n^2 + n - (2*n^3 - 7*n^2 + 9*n)*x + (2*n^3 - 7*n^2 + 5*n - 6)*x^2 + (n^2 - 5*n + 6)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n^2 - n + (2*n^3 + n^2 - 7*n)*x + (6*n^3 - 11*n^2 + 5*n - 6)*x^2 + (n^2 - 5*n + 6)*x^3)*v[i - 1] + (n*x + (2*n^2 + 6*n + 1)*x^2 + (12*n^2 + 2*n + 5)*x^3 + (2*n^2 + 6*n - 5)*x^4 + (n - 1)*x^5)*deriv(v[i - 1], x))/(-n^2 + n - (2*n^3 - 7*n^2 + 9*n)*x + (2*n^3 - 7*n^2 + 5*n - 6)*x^2 + (n^2 - 5*n + 6)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 3
  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 Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0…?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
        • 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), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Liu–Wang theorem with a lag row (above)

Related ?

Same fingerprint ?

• A266213 → θ+1

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.