A181543 → D+1

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Triangle begins:
  0:   1;
  1:   2,     1;
  2:   9,    10,       1;
  3:  28,    81,      30,       1;
  4:  65,   496,     408,      68,        1;
  5: 126,  2125,    4000,    1500,      130,       1;
  6: 217,  6966,   27375,   21500,     4455,     222,      1;
  7: 344, 18865,  137886,  214375,    89180,   11319,    350,     1;
  8: 513, 44416,  548800, 1547616,  1221080,  307328,  25536,   520,   1;
  9: 730, 94041, 1824768, 8594208, 12002256, 5556600, 919296, 52488, 738, 1;

The row polynomials satisfy (n^5-3*n^4+3*n^3+(3*n^4-9*n^3+4*n^2)*x+(3*n^3-5*n^2)*x^2+n^2*x^3)*P(n,x) = (n^5+n^2+(10*n^5-42*n^4+51*n^3-28*n^2+6*n)*x+(30*n^4-120*n^3+147*n^2-75*n+12)*x^2+(30*n^3-84*n^2+78*n-24)*x^3+(10*n^2-15*n+6)*x^4)*P(n-1,x) + ((3*n^4+9*n^3-15*n^2)*x-(15*n^4-60*n^3+22*n^2+42*n)*x^2-(45*n^3-152*n^2+105*n+12)*x^3-(45*n^2-102*n+48)*x^4-(15*n-18)*x^5)*d/dx P(n-1,x) + ((6*n^3-8*n^2)*x^2+(6*n^3+10*n^2-24*n)*x^3+(18*n^2-6*n-12)*x^4+(18*n-6)*x^5+6*x^6)*d^2/dx^2 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^5 + n^2 + (10*n^5 - 42*n^4 + 51*n^3 - 28*n^2 + 6*n)*x + (30*n^4 - 120*n^3 + 147*n^2 - 75*n + 12)*x^2 + (30*n^3 - 84*n^2 + 78*n - 24)*x^3 + (10*n^2 - 15*n + 6)*x^4)*P[n - 1, x] + ((3*n^4 + 9*n^3 - 15*n^2)*x - (15*n^4 - 60*n^3 + 22*n^2 + 42*n)*x^2 - (45*n^3 - 152*n^2 + 105*n + 12)*x^3 - (45*n^2 - 102*n + 48)*x^4 - (15*n - 18)*x^5)*D[P[n - 1, x], x] + ((6*n^3 - 8*n^2)*x^2 + (6*n^3 + 10*n^2 - 24*n)*x^3 + (18*n^2 - 6*n - 12)*x^4 + (18*n - 6)*x^5 + 6*x^6)*D[P[n - 1, x], {x, 2}])/(n^5 - 3*n^4 + 3*n^3 + (3*n^4 - 9*n^3 + 4*n^2)*x + (3*n^3 - 5*n^2)*x^2 + n^2*x^3)]];
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^5 + n^2 + (10*n^5 - 42*n^4 + 51*n^3 - 28*n^2 + 6*n)*x + (30*n^4 - 120*n^3 + 147*n^2 - 75*n + 12)*x^2 + (30*n^3 - 84*n^2 + 78*n - 24)*x^3 + (10*n^2 - 15*n + 6)*x^4)*v[i - 1] + ((3*n^4 + 9*n^3 - 15*n^2)*x - (15*n^4 - 60*n^3 + 22*n^2 + 42*n)*x^2 - (45*n^3 - 152*n^2 + 105*n + 12)*x^3 - (45*n^2 - 102*n + 48)*x^4 - (15*n - 18)*x^5)*deriv(v[i - 1], x) + ((6*n^3 - 8*n^2)*x^2 + (6*n^3 + 10*n^2 - 24*n)*x^3 + (18*n^2 - 6*n - 12)*x^4 + (18*n - 6)*x^5 + 6*x^6)*deriv(deriv(v[i - 1], x), x))/(n^5 - 3*n^4 + 3*n^3 + (3*n^4 - 9*n^3 + 4*n^2)*x + (3*n^3 - 5*n^2)*x^2 + n^2*x^3)); 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

Proof chains ? JSON

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

  • from Triangle of cubed binomial coefficients, T(n,k) = C(n,k)^3, read by rows.?
    • Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Square the entries of Pascal's triangle.?
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? 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
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Square the entries of Pascal's triangle.?
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Three-term recurrence criterion (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Square the entries of Pascal's triangle.?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Three-term recurrence criterion (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle of cubed binomial coefficients, T(n,k) = C(n,k)^3, read by rows.?
        • ↑ Pólya–Schur theorem on multiplier sequences (above)
        • 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
    • ↑ Row-hypergeometric rows (multiplier sequences) (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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
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.
5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023