A112007 → θ+1

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Submission text ?

Triangle begins:
  0:       1;
  1:       2,        2;
  2:       6,       16,         3;
  3:      24,      116,        66,         4;
  4:     120,      888,       984,       208,         5;
  5:     720,     7416,     13200,      5808,       570,         6;
  6:    5040,    67968,    174420,    128480,     28050,      1440,        7;
  7:   40320,   682272,   2355912,   2576080,    979000,    119700,     3458,       8;
  8:  362880,  7467840,  33078888,  49760256,  28827500,   6375000,   470820,    8032,     9;
  9: 3628800, 88678080, 486560736, 955619616, 776786920, 268590000, 37283120, 1750784, 18234, 10;

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

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n + (2*n^3 - 2*n^2 - 2*n + 2)*x + (n^3 - n)*x^2)*P[n - 2, x] - ((n^2 - 1)*x + (n + 1)*x^2 - (n^2 + n)*x^3)*D[P[n - 2, x], x] + (2*n^2 + n - 1 + 2*n^2*x)*P[n - 1, x] + (n*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] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - n + (2*n^3 - 2*n^2 - 2*n + 2)*x + (n^3 - n)*x^2)*v[i - 2] - ((n^2 - 1)*x + (n + 1)*x^2 - (n^2 + n)*x^3)*deriv(v[i - 2], x) + (2*n^2 + n - 1 + 2*n^2*x)*v[i - 1] + (n*x - 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 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 [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 (above)

Related ?

Same fingerprint ?

• A163936 → θ+1
• A201637 → reverse (shift +1) → D

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.