A225472 → (θ+1)²

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Submission text ?

Triangle begins:
   1:   1;
   2:   2,      12;
   3:   4,      84,        162;
   4:   8,     468,       2430,        2592;
   5:  16,    2436,      25758,       67392,        48600;
   6:  32,   12372,     238950,     1153440,      1944000,       1049760;
   7:  64,   62244,    2078622,    16511040,     48843000,      59836320,      25719120;
   8: 128,  311988,   17469270,   214879392,    993384000,    2072226240,    1980372240,     705438720;
   9: 256, 1561476,  143965998,  2643181632,  17936364600,   56684940480,   90376987680,   70543872000,   21427701120;
  10: 512, 7810452, 1172807910, 31378453920, 300668760000, 1351069463520, 3196320795360, 4101420718080, 2699890341120, 714256704000;

The row polynomials satisfy P(n,x) = (2+12*x-18*x^2)*P(n-2,x) + (3*x-6*x^2-9*x^3)*d/dx P(n-2,x) + (1+12*x)*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) = 2 + 12*x.

P[1, x_] := 1;
P[2, x_] := 2 + 12*x;
P[n_, x_] := P[n, x] = Expand[(2 + 12*x - 18*x^2)*P[n - 2, x] + (3*x - 6*x^2 - 9*x^3)*D[P[n - 2, x], x] + (1 + 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] = 1; v[2] = 2 + 12*x; for(n = 3, N, my(i = n); v[i] = (2 + 12*x - 18*x^2)*v[i - 2] + (3*x - 6*x^2 - 9*x^3)*deriv(v[i - 2], x) + (1 + 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from 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 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.