A114655 → x ↦ x+1 → θ+1

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Triangle begins:
   1:       2;
   2:       6,       4;
   3:      22,      32,        6;
   4:      90,     204,       90,        8;
   5:     394,    1216,      876,      192,      10;
   6:    1806,    7060,     7140,     2640,     350,      12;
   7:    8558,   40512,    53010,    28160,    6450,     576,     14;
   8:   41586,  231196,   372078,   260120,   87150,   13692,    882,    16;
   9:  206098, 1316096,  2517144,  2192512,  980140,  228480,  26264,  1280,   18;
  10: 1037718, 7484580, 16596360, 17354400, 9771300, 3073896, 530376, 46656, 1782, 20;

The row polynomials satisfy -(2*n+2-(2*n^2+2*n)*x-(n^3-n)*x^2)*P(n,x) = -(10*n-2-(8*n^2-8*n+8)*x-(5*n^3+6*n^2-11*n)*x^2-(3*n^3-2*n^2-3*n+2)*x^3)*P(n-1,x) - (2-4*n*x-(2*n^2+n+1)*x^2+4*n*x^3+(2*n^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[(-(10*n - 2 - (8*n^2 - 8*n + 8)*x - (5*n^3 + 6*n^2 - 11*n)*x^2 - (3*n^3 - 2*n^2 - 3*n + 2)*x^3)*P[n - 1, x] - (2 - 4*n*x - (2*n^2 + n + 1)*x^2 + 4*n*x^3 + (2*n^2 + n - 1)*x^4)*D[P[n - 1, x], x])/(-2*n - 2 + (2*n^2 + 2*n)*x + (n^3 - 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] = (-(10*n - 2 - (8*n^2 - 8*n + 8)*x - (5*n^3 + 6*n^2 - 11*n)*x^2 - (3*n^3 - 2*n^2 - 3*n + 2)*x^3)*v[i - 1] - (2 - 4*n*x - (2*n^2 + n + 1)*x^2 + 4*n*x^3 + (2*n^2 + n - 1)*x^4)*deriv(v[i - 1], x))/(-2*n - 2 + (2*n^2 + 2*n)*x + (n^3 - 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 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

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from shift_x_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascen…?
    • 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 [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)
    • 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) (above)

Related ?

Same fingerprint ?

• A114656 → basis A007318 → θ+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.