A008517 → θ+1

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Submission text ?

Triangle begins:
  0: 1;
  1: 1,    4;
  2: 1,   16,     18;
  3: 1,   44,    174,       96;
  4: 1,  104,    984,     1776,       600;
  5: 1,  228,   4356,    17600,     18540,      4320;
  6: 1,  480,  16830,   128480,    290700,    203904,      35280;
  7: 1,  988,  59850,   783200,   3220100,   4711824,    2387952,     322560;
  8: 1, 2008, 201780,  4250000,  28827500,  74640384,   77184072,   29871360,   3265920;
  9: 1, 4052, 656544, 21304640, 223825000, 932144304, 1672334328, 1297495296, 399051360, 36288000;

The row polynomials satisfy P(n,x) = -(4*n^2-2*n-2)*x^2*P(n-2,x) - ((2*n+1)*x^2-(2*n+1)*x^3)*d/dx P(n-2,x) + (1+4*n*x)*P(n-1,x) + (x-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[-(4*n^2 - 2*n - 2)*x^2*P[n - 2, x] - ((2*n + 1)*x^2 - (2*n + 1)*x^3)*D[P[n - 2, x], x] + (1 + 4*n*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
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] = -(4*n^2 - 2*n - 2)*x^2*v[i - 2] - ((2*n + 1)*x^2 - (2*n + 1)*x^3)*deriv(v[i - 2], x) + (1 + 4*n*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); 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 Second-order Eulerian triangle T(n,k), 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 Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Second-order Eulerian triangle T(n,k), 1 <= k <= n.?
            • ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

• A163936 → reverse → D
• A201637 → 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.