A114656 → reverse → (θ+1)²

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Triangle begins:
   1: 1;
   2: 1,   8;
   3: 1,  24,    36;
   4: 1,  48,   216,    128;
   5: 1,  80,   720,   1280,     400;
   6: 1, 120,  1800,   6400,    6000,    1152;
   7: 1, 168,  3780,  22400,   42000,   24192,    3136;
   8: 1, 224,  7056,  62720,  196000,  225792,   87808,    8192;
   9: 1, 288, 12096, 150528,  705600, 1354752, 1053696,  294912,  20736;
  10: 1, 360, 19440, 322560, 2116800, 6096384, 7902720, 4423680, 933120, 51200;

The row polynomials satisfy -(n^2-n-(2*n^4-4*n^3+2*n^2)*x)*P(n,x) = -(n^2-n-(2*n^4-4*n^3-10*n^2+20*n-8)*x-(12*n^4-16*n^3-8*n^2+20*n-8)*x^2)*P(n-1,x) - ((2*n-1)*x-4*n^3*x^2+(8*n^3-8*n+4)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n - (2*n^4 - 4*n^3 - 10*n^2 + 20*n - 8)*x - (12*n^4 - 16*n^3 - 8*n^2 + 20*n - 8)*x^2)*P[n - 1, x] - ((2*n - 1)*x - 4*n^3*x^2 + (8*n^3 - 8*n + 4)*x^3)*D[P[n - 1, x], x])/(-n^2 + n + (2*n^4 - 4*n^3 + 2*n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n^2 - n - (2*n^4 - 4*n^3 - 10*n^2 + 20*n - 8)*x - (12*n^4 - 16*n^3 - 8*n^2 + 20*n - 8)*x^2)*v[i - 1] - ((2*n - 1)*x - 4*n^3*x^2 + (8*n^3 - 8*n + 4)*x^3)*deriv(v[i - 1], x))/(-n^2 + n + (2*n^4 - 4*n^3 + 2*n^2)*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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by 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 reversal of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two c…?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
        • 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 reversal of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two c…?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also …?
                • ↑ 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)

Related ?

Same fingerprint ?

• A114687 → (θ+1)²
• A114655 → reverse → (θ+1)²

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