A001263 → basis A013609 → θ+1

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Triangle begins:
  0:     1;
  1:     2,      4;
  2:     5,     20,      12;
  3:    14,     84,     108,      32;
  4:    42,    336,     672,     448,      80;
  5:   132,   1320,    3600,    3840,    1600,     192;
  6:   429,   5148,   17820,   26400,   18000,    5184,     448;
  7:  1430,  20020,   84084,  160160,  154000,   73920,   15680,   1024;
  8:  4862,  77792,  384384,  896896, 1121120,  768768,  275968,  45056,   2304;
  9: 16796, 302328, 1718496, 4752384, 7338240, 6604416, 3424512, 958464, 124416, 5120;

The row polynomials satisfy (n^2+3*n+2+(n^3+3*n^2+2*n)*x)*P(n,x) = (4*n^2+6*n+2+(4*n^3+18*n^2+14*n)*x+(6*n^3+14*n^2+4*n)*x^2)*P(n-1,x) - ((2*n+2)*x+(2*n^2+9*n+6)*x^2+(4*n^2+10*n+4)*x^3)*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[((4*n^2 + 6*n + 2 + (4*n^3 + 18*n^2 + 14*n)*x + (6*n^3 + 14*n^2 + 4*n)*x^2)*P[n - 1, x] - ((2*n + 2)*x + (2*n^2 + 9*n + 6)*x^2 + (4*n^2 + 10*n + 4)*x^3)*D[P[n - 1, x], x])/(n^2 + 3*n + 2 + (n^3 + 3*n^2 + 2*n)*x)]];
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] = ((4*n^2 + 6*n + 2 + (4*n^3 + 18*n^2 + 14*n)*x + (6*n^3 + 14*n^2 + 4*n)*x^2)*v[i - 1] - ((2*n + 2)*x + (2*n^2 + 9*n + 6)*x^2 + (4*n^2 + 10*n + 4)*x^3)*deriv(v[i - 1], x))/(n^2 + 3*n + 2 + (n^3 + 3*n^2 + 2*n)*x)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
  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 basis_map_A013609 of Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also c…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…?
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Three-term recurrence criterion [Sze39, §3.3; Fav35] 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…?
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • ↑ Three-term recurrence criterion (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• narayana → basis A013609 → 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. [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