A062190 → reverse → D+1

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Triangle begins:
  0:     1;
  1:     7,      1;
  2:    35,     16,      1;
  3:   140,    132,     27,       1;
  4:   462,    768,    324,      40,      1;
  5:  1302,   3450,   2550,     650,     55,      1;
  6:  3234,  12672,  14850,    6600,   1155,     72,     1;
  7:  7260,  39732,  68607,   48125,  14630,   1890,    91,    1;
  8: 15015, 109824, 264264,  272272, 130130,  29120,  2912,  112,   1;
  9: 29029, 274131, 880308, 1261260, 882882, 309582, 53508, 4284, 135, 1;

The row polynomials satisfy (n^4+3*n^3-3*n^2+35*n+(2*n^3+6*n^2-20*n)*x+(n^2+5*n)*x^2)*P(n,x) = (n^4+10*n^3+31*n^2+60*n+150+(3*n^4+6*n^3+18*n^2+50*n-125)*x+(6*n^3+8*n^2-24*n+40)*x^2+(3*n^2+8*n-5)*x^3)*P(n-1,x) + ((2*n^3+4*n^2+10*n+25)*x-(2*n^3+4*n+35)*x^2-(4*n^2+4*n-15)*x^3-(2*n+5)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^4 + 10*n^3 + 31*n^2 + 60*n + 150 + (3*n^4 + 6*n^3 + 18*n^2 + 50*n - 125)*x + (6*n^3 + 8*n^2 - 24*n + 40)*x^2 + (3*n^2 + 8*n - 5)*x^3)*P[n - 1, x] + ((2*n^3 + 4*n^2 + 10*n + 25)*x - (2*n^3 + 4*n + 35)*x^2 - (4*n^2 + 4*n - 15)*x^3 - (2*n + 5)*x^4)*D[P[n - 1, x], x])/(n^4 + 3*n^3 - 3*n^2 + 35*n + (2*n^3 + 6*n^2 - 20*n)*x + (n^2 + 5*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^4 + 10*n^3 + 31*n^2 + 60*n + 150 + (3*n^4 + 6*n^3 + 18*n^2 + 50*n - 125)*x + (6*n^3 + 8*n^2 - 24*n + 40)*x^2 + (3*n^2 + 8*n - 5)*x^3)*v[i - 1] + ((2*n^3 + 4*n^2 + 10*n + 25)*x - (2*n^3 + 4*n + 35)*x^2 - (4*n^2 + 4*n - 15)*x^3 - (2*n + 5)*x^4)*deriv(v[i - 1], x))/(n^4 + 3*n^3 - 3*n^2 + 35*n + (2*n^3 + 6*n^2 - 20*n)*x + (n^2 + 5*n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

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

  • from reversal of Coefficient triangle of certain polynomials N(5; m,x).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Coefficient triangle of certain polynomials N(5; m,x).?
        • 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
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Coefficient triangle of certain polynomials N(5; m,x).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Coefficient triangle of certain polynomials N(5; m,x).?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (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.