A062190 → reverse → D

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Triangle begins:
  0:     0;
  1:     1;
  2:    14,      2;
  3:    84,     48,      3;
  4:   336,    432,    108,       4;
  5:  1050,   2400,   1350,     200,      5;
  6:  2772,   9900,   9900,    3300,    330,      6;
  7:  6468,  33264,  51975,   30800,   6930,    504,     7;
  8: 13728,  96096, 216216,  200200,  80080,  13104,   728,    8;
  9: 27027, 247104, 756756, 1009008, 630630, 183456, 22932, 1008, 9;

The row polynomials satisfy (n^3+3*n^2-9*n+5)*P(n,x) = (n^3+10*n^2+25*n+(3*n^3+4*n^2-13*n)*x)*P(n-1,x) + ((2*n^2+4*n)*x-(2*n^2+4*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 10*n^2 + 25*n + (3*n^3 + 4*n^2 - 13*n)*x)*P[n - 1, x] + ((2*n^2 + 4*n)*x - (2*n^2 + 4*n)*x^2)*D[P[n - 1, x], x])/(n^3 + 3*n^2 - 9*n + 5)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 + 10*n^2 + 25*n + (3*n^3 + 4*n^2 - 13*n)*x)*v[i - 1] + ((2*n^2 + 4*n)*x - (2*n^2 + 4*n)*x^2)*deriv(v[i - 1], x))/(n^3 + 3*n^2 - 9*n + 5)); 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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.