A114655 → θ+1 → D+1

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Triangle begins:
   1:     2;
   2:     8,      4;
   3:    32,     36,       6;
   4:   112,    240,      96,       8;
   5:   352,   1280,     960,     200,      10;
   6:  1024,   5760,    7200,    2800,     360,      12;
   7:  2816,  22848,   43680,   28000,    6720,     588,     14;
   8:  7424,  82432,  225792,  219520,   86240,   14112,    896,    16;
   9: 18944, 276480, 1032192, 1430016,  846720,  225792,  26880,  1296,   18;
  10: 47104, 875520, 4285440, 8064000, 6773760, 2709504, 524160, 47520, 1800, 20;

The row polynomials satisfy (n^4-4*n^3+7*n^2-4*n+(2*n^3-8*n^2+6*n)*x+(n^2-n)*x^2)*P(n,x) = (2*n^4-4*n^3+6*n^2-4*n+(3*n^4-8*n^3-n^2+6*n)*x+(6*n^3-23*n^2+27*n-10)*x^2+(3*n^2-5*n+2)*x^3)*P(n-1,x) + ((4*n^3-8*n^2+4*n)*x-(2*n^3-12*n^2+22*n-8)*x^2-(4*n^2-14*n+6)*x^3-(2*n-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.

P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^4 - 4*n^3 + 6*n^2 - 4*n + (3*n^4 - 8*n^3 - n^2 + 6*n)*x + (6*n^3 - 23*n^2 + 27*n - 10)*x^2 + (3*n^2 - 5*n + 2)*x^3)*P[n - 1, x] + ((4*n^3 - 8*n^2 + 4*n)*x - (2*n^3 - 12*n^2 + 22*n - 8)*x^2 - (4*n^2 - 14*n + 6)*x^3 - (2*n - 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 4*n^3 + 7*n^2 - 4*n + (2*n^3 - 8*n^2 + 6*n)*x + (n^2 - n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((2*n^4 - 4*n^3 + 6*n^2 - 4*n + (3*n^4 - 8*n^3 - n^2 + 6*n)*x + (6*n^3 - 23*n^2 + 27*n - 10)*x^2 + (3*n^2 - 5*n + 2)*x^3)*v[i - 1] + ((4*n^3 - 8*n^2 + 4*n)*x - (2*n^3 - 12*n^2 + 22*n - 8)*x^2 - (4*n^2 - 14*n + 6)*x^3 - (2*n - 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 4*n^3 + 7*n^2 - 4*n + (2*n^3 - 8*n^2 + 6*n)*x + (n^2 - n)*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
• 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 theta_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schr…?
        • 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) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k…?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schr…?
                • ↑ 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) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schr…?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schr…?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A114656 → θ+1 → D+1

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.