A062145 → D → D+1

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Triangle begins:
   1:    0;
   2:    4;
   3:   30,     20;
   4:  108,    210,     60;
   5:  280,   1092,    840,     140;
   6:  600,   3920,   5880,    2520,     280;
   7: 1134,  11160,  27720,   22680,    6300,     504;
   8: 1960,  27090, 100800,  135240,   70560,   13860,    840;
   9: 3168,  58520, 304920,  609840,  517440,  188496,  27720,  1320;
  10: 4860, 115632, 803880, 2245320, 2827440, 1663200, 449064, 51480, 1980;

The row polynomials satisfy (n^4-4*n^3+3*n^2+4*n-4+(2*n^3-7*n^2+n+10)*x+(n^2-n-2)*x^2)*P(n,x) = (n^4-3*n^2+2*n+(3*n^4-8*n^3+5*n^2-2*n+4)*x+(6*n^3-17*n^2+17*n-10)*x^2+(3*n^2-3*n+2)*x^3)*P(n-1,x) + ((2*n^3-3*n^2-n+2)*x-(2*n^3-7*n^2+5*n+2)*x^2-(4*n^2-8*n)*x^3-2*n*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.

P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((n^4 - 3*n^2 + 2*n + (3*n^4 - 8*n^3 + 5*n^2 - 2*n + 4)*x + (6*n^3 - 17*n^2 + 17*n - 10)*x^2 + (3*n^2 - 3*n + 2)*x^3)*P[n - 1, x] + ((2*n^3 - 3*n^2 - n + 2)*x - (2*n^3 - 7*n^2 + 5*n + 2)*x^2 - (4*n^2 - 8*n)*x^3 - 2*n*x^4)*D[P[n - 1, x], x])/(n^4 - 4*n^3 + 3*n^2 + 4*n - 4 + (2*n^3 - 7*n^2 + n + 10)*x + (n^2 - n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = ((n^4 - 3*n^2 + 2*n + (3*n^4 - 8*n^3 + 5*n^2 - 2*n + 4)*x + (6*n^3 - 17*n^2 + 17*n - 10)*x^2 + (3*n^2 - 3*n + 2)*x^3)*v[i - 1] + ((2*n^3 - 3*n^2 - n + 2)*x - (2*n^3 - 7*n^2 + 5*n + 2)*x^2 - (4*n^2 - 8*n)*x^3 - 2*n*x^4)*deriv(v[i - 1], x))/(n^4 - 4*n^3 + 3*n^2 + 4*n - 4 + (2*n^3 - 7*n^2 + n + 10)*x + (n^2 - n - 2)*x^2)); 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 derivative of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n…?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n -…?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) *…?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (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.