A039755 → reverse → D

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Submission text ?

Triangle begins:
   1:  0;
   2:  1;
   3:  4,    2;
   4:  9,   26,     3;
   5: 16,  116,   120,      4;
   6: 25,  340,   990,    484,       5;
   7: 36,  790,  4560,   7084,    1820,       6;
   8: 49, 1582, 15225,  49644,   46095,    6558,       7;
   9: 64, 2856, 41328, 232344,  480480,  283128,   22960,     8;
  10: 81, 4776, 97020, 838488, 3094350, 4319160, 1674540, 78728, 9;

The row polynomials satisfy (2*n-5)*P(n,x) = -(2*n-3+(8*n^2-40*n+42)*x+(8*n^3-60*n^2+142*n-105)*x^2)*P(n-2,x) + ((4*n-6)*x^2+(8*n^2-32*n+30)*x^3)*d/dx P(n-2,x) + (4*n-8+(8*n^2-40*n+50)*x)*P(n-1,x) - (4*n-10)*x^2*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 3 + (8*n^2 - 40*n + 42)*x + (8*n^3 - 60*n^2 + 142*n - 105)*x^2)*P[n - 2, x] + ((4*n - 6)*x^2 + (8*n^2 - 32*n + 30)*x^3)*D[P[n - 2, x], x] + (4*n - 8 + (8*n^2 - 40*n + 50)*x)*P[n - 1, x] - (4*n - 10)*x^2*D[P[n - 1, x], x])/(2*n - 5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n - 3 + (8*n^2 - 40*n + 42)*x + (8*n^3 - 60*n^2 + 142*n - 105)*x^2)*v[i - 2] + ((4*n - 6)*x^2 + (8*n^2 - 32*n + 30)*x^3)*deriv(v[i - 2], x) + (4*n - 8 + (8*n^2 - 40*n + 50)*x)*v[i - 1] - (4*n - 10)*x^2*deriv(v[i - 1], x))/(2*n - 5)); 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

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

  • from reversal of Triangle of B-analogs of Stirling numbers of the second kind.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle of B-analogs of Stirling numbers of the second kind.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle of B-analogs of Stirling numbers of the second kind.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of B-analogs of Stirling numbers of the second kind.?
                • ↑ Liu–Wang theorem (above)
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • ↑ Iterated real-rootedness preservers (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A039756 → D

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.