A062190 → reverse → θ+1

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Triangle begins:
  0:    1;
  1:    6,     2;
  2:   21,    28,      3;
  3:   56,   168,     72,       4;
  4:  126,   672,    648,     144,       5;
  5:  252,  2100,   3600,    1800,     250,      6;
  6:  462,  5544,  14850,   13200,    4125,    396,      7;
  7:  792, 12936,  49896,   69300,   38500,   8316,    588,     8;
  8: 1287, 27456, 144144,  288288,  250250,  96096,  15288,   832,    9;
  9: 2002, 54054, 370656, 1009008, 1261260, 756756, 214032, 26208, 1134, 10;

The row polynomials satisfy (4*n^2+20*n+(n^4+5*n^3)*x)*P(n,x) = (4*n^2+40*n+100+(n^4+10*n^3+49*n^2+64*n-40)*x+(3*n^4+12*n^3+7*n^2-10*n)*x^2)*P(n-1,x) + ((8*n+20)*x+(2*n^3+8*n^2+2*n-20)*x^2-(2*n^3+8*n^2+10*n)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 40*n + 100 + (n^4 + 10*n^3 + 49*n^2 + 64*n - 40)*x + (3*n^4 + 12*n^3 + 7*n^2 - 10*n)*x^2)*P[n - 1, x] + ((8*n + 20)*x + (2*n^3 + 8*n^2 + 2*n - 20)*x^2 - (2*n^3 + 8*n^2 + 10*n)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 20*n + (n^4 + 5*n^3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 40*n + 100 + (n^4 + 10*n^3 + 49*n^2 + 64*n - 40)*x + (3*n^4 + 12*n^3 + 7*n^2 - 10*n)*x^2)*v[i - 1] + ((8*n + 20)*x + (2*n^3 + 8*n^2 + 2*n - 20)*x^2 - (2*n^3 + 8*n^2 + 10*n)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 20*n + (n^4 + 5*n^3)*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 reversal of Coefficient triangle of certain polynomials N(5; m,x).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Coefficient triangle of certain polynomials N(5; m,x).?
        • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Coefficient triangle of certain polynomials N(5; m,x).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Coefficient triangle of certain polynomials N(5; m,x).?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (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.