A225472 → D+1

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Submission text ?

Triangle begins:
   1:       1;
   2:       5,         3;
   3:      25,        57,         18;
   4:     125,       657,        756,         162;
   5:     625,      6333,      15498,       11988,         1944;
   6:    3125,     56193,     242820,      383130,       223560,        29160;
   7:   15625,    477477,    3326778,     8846820,     10264320,      4811400,       524880;
   8:   78125,   3960057,   42230916,   172371402,    327544560,    300056400,    117573120,     11022480;
   9:  390625,  32382813,  511592778,  3035017188,   8590362984,  12641151600,   9560164320,   3218564160,   264539520;
  10: 1953125, 262576593, 6013772100, 50068154970, 199675287000, 428915927160, 513823927680, 330740534880, 97615082880, 7142567040;

The row polynomials satisfy P(n,x) = -(10+18*x+3*x^2-9*x^3)*P(n-2,x) - (6*x+6*x^2-9*x^3-9*x^4)*d/dx P(n-2,x) + (7+9*x-3*x^2)*P(n-1,x) + (3*x+3*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 3*x.

P[1, x_] := 1;
P[2, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(10 + 18*x + 3*x^2 - 9*x^3)*P[n - 2, x] - (6*x + 6*x^2 - 9*x^3 - 9*x^4)*D[P[n - 2, x], x] + (7 + 9*x - 3*x^2)*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] = 1; v[2] = 5 + 3*x; for(n = 3, N, my(i = n); v[i] = -(10 + 18*x + 3*x^2 - 9*x^3)*v[i - 2] - (6*x + 6*x^2 - 9*x^3 - 9*x^4)*deriv(v[i - 2], x) + (7 + 9*x - 3*x^2)*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.