A001263 → D → D+1

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Triangle begins:
  0:    0;
  1:    1;
  2:    5,     2;
  3:   18,    18,     3;
  4:   50,   100,    42,      4;
  5:  115,   400,   330,     80,      5;
  6:  231,  1260,  1785,    840,    135,     6;
  7:  420,  3332,  7350,   5880,   1820,   210,    7;
  8:  708,  7728, 24696,  30576,  15960,  3528,  308,   8;
  9: 1125, 16200, 71064, 127008, 102060, 37800, 6300, 432, 9;

The row polynomials satisfy (n^4-2*n^3+11*n-10+(2*n^3-3*n^2-9*n+10)*x+(n^2+n-2)*x^2)*P(n,x) = (n^4+2*n^3+2*n^2+7*n+6+(3*n^4-6*n^3+10*n^2-11*n-14)*x+(6*n^3-13*n^2+3*n+10)*x^2+(3*n^2-n-2)*x^3)*P(n-1,x) + ((2*n^3-2*n^2+4*n+2)*x-(2*n^3-6*n^2+8*n+5)*x^2-(4*n^2-6*n-4)*x^3-(2*n+1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^4 + 2*n^3 + 2*n^2 + 7*n + 6 + (3*n^4 - 6*n^3 + 10*n^2 - 11*n - 14)*x + (6*n^3 - 13*n^2 + 3*n + 10)*x^2 + (3*n^2 - n - 2)*x^3)*P[n - 1, x] + ((2*n^3 - 2*n^2 + 4*n + 2)*x - (2*n^3 - 6*n^2 + 8*n + 5)*x^2 - (4*n^2 - 6*n - 4)*x^3 - (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 2*n^3 + 11*n - 10 + (2*n^3 - 3*n^2 - 9*n + 10)*x + (n^2 + n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^4 + 2*n^3 + 2*n^2 + 7*n + 6 + (3*n^4 - 6*n^3 + 10*n^2 - 11*n - 14)*x + (6*n^3 - 13*n^2 + 3*n + 10)*x^2 + (3*n^2 - n - 2)*x^3)*v[i - 1] + ((2*n^3 - 2*n^2 + 4*n + 2)*x - (2*n^3 - 6*n^2 + 8*n + 5)*x^2 - (4*n^2 - 6*n - 4)*x^3 - (2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 2*n^3 + 11*n - 10 + (2*n^3 - 3*n^2 - 9*n + 10)*x + (n^2 + n - 2)*x^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

Proof chains ? JSON

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

  • from derivative 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 called t…?
    • Laguerre’s theorem on differential operators [SP14] 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)
    • Laguerre’s theorem on differential operators [SP14] 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 → D → D+1
• A352880 → reverse → D+1
• A108838 → 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. [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