A001263 → basis A013609 → D

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Triangle begins:
  0:      0;
  1:      2;
  2:     10,       8;
  3:     42,      72,      24;
  4:    168,     448,     336,      64;
  5:    660,    2400,    2880,    1280,     160;
  6:   2574,   11880,   19800,   14400,    4320,     384;
  7:  10010,   56056,  120120,  123200,   61600,   13440,    896;
  8:  38896,  256256,  672672,  896896,  640640,  236544,  39424,   2048;
  9: 151164, 1145664, 3564288, 5870592, 5503680, 2935296, 838656, 110592, 4608;

The row polynomials satisfy (n^2+n-2)*P(n,x) = (4*n^2+2*n+(6*n^2-2*n-4)*x)*P(n-1,x) - ((2*n+1)*x+(4*n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 2*n + (6*n^2 - 2*n - 4)*x)*P[n - 1, x] - ((2*n + 1)*x + (4*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 + 2*n + (6*n^2 - 2*n - 4)*x)*v[i - 1] - ((2*n + 1)*x + (4*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + n - 2)); 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

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

• A352880 → reverse → basis A013609
• narayana → basis A013609 → D
• A108838 → basis A013609

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