A225472 → D

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Triangle begins:
   1:       0;
   2:       3;
   3:      21,        36;
   4:     117,       540,        486;
   5:     609,      5724,      12636,        7776;
   6:    3093,     53100,     216270,      311040,       145800;
   7:   15561,    461916,    3095820,     7814880,      8310600,      3149280;
   8:   77997,   3882060,   40289886,   158941440,    287809200,    242494560,     77157360;
   9:  390369,  31992444,  495596556,  2869818336,   7872908400,  11066569920,   7715736000,   2116316160;
  10: 1952613, 260623980, 5883460110, 48107001600, 187648536600, 391386219840, 448592891040, 266655836160, 64283103360;

The row polynomials satisfy P(n,x) = -(10+24*x+18*x^2)*P(n-2,x) - (6*x+15*x^2+9*x^3)*d/dx P(n-2,x) + (7+12*x)*P(n-1,x) + (3*x+3*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.

P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[-(10 + 24*x + 18*x^2)*P[n - 2, x] - (6*x + 15*x^2 + 9*x^3)*D[P[n - 2, x], x] + (7 + 12*x)*P[n - 1, x] + (3*x + 3*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = -(10 + 24*x + 18*x^2)*v[i - 2] - (6*x + 15*x^2 + 9*x^3)*deriv(v[i - 2], x) + (7 + 12*x)*v[i - 1] + (3*x + 3*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, 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= …?
    • Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
    • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [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
4. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.