A001263 → D → (θ+1)²

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Submission text ?

Triangle begins:
  0:  0;
  1:  1;
  2:  3,    8;
  3:  6,   48,    27;
  4: 10,  160,   270,     64;
  5: 15,  400,  1350,    960,    125;
  6: 21,  840,  4725,   6720,   2625,    216;
  7: 28, 1568, 13230,  31360,  24500,   6048,    343;
  8: 36, 2688, 31752, 112896, 147000,  72576,  12348,   512;
  9: 45, 4320, 68040, 338688, 661500, 544320, 185220, 23040, 729;

The row polynomials satisfy (n^2+n-2+(2*n^4-2*n^3-7*n^2+9*n-2)*x+(n^6-n^5-3*n^4+5*n^3-2*n^2)*x^2)*P(n,x) = (n^2+3*n+2+(2*n^4+2*n^3+5*n^2-9*n-6)*x+(n^6+n^5+6*n^4-13*n^3-6*n^2+5*n+6)*x^2+(3*n^6+n^5-6*n^4-3*n^3+6*n^2+n-2)*x^3)*P(n-1,x) + ((2*n+1)*x+(4*n^3-4*n-3)*x^2+(2*n^5+3*n^4-4*n^3-3*n^2+2*n+3)*x^3-(2*n^5+3*n^4-3*n^2+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^2 + 3*n + 2 + (2*n^4 + 2*n^3 + 5*n^2 - 9*n - 6)*x + (n^6 + n^5 + 6*n^4 - 13*n^3 - 6*n^2 + 5*n + 6)*x^2 + (3*n^6 + n^5 - 6*n^4 - 3*n^3 + 6*n^2 + n - 2)*x^3)*P[n - 1, x] + ((2*n + 1)*x + (4*n^3 - 4*n - 3)*x^2 + (2*n^5 + 3*n^4 - 4*n^3 - 3*n^2 + 2*n + 3)*x^3 - (2*n^5 + 3*n^4 - 3*n^2 + 1)*x^4)*D[P[n - 1, x], x])/(n^2 + n - 2 + (2*n^4 - 2*n^3 - 7*n^2 + 9*n - 2)*x + (n^6 - n^5 - 3*n^4 + 5*n^3 - 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^2 + 3*n + 2 + (2*n^4 + 2*n^3 + 5*n^2 - 9*n - 6)*x + (n^6 + n^5 + 6*n^4 - 13*n^3 - 6*n^2 + 5*n + 6)*x^2 + (3*n^6 + n^5 - 6*n^4 - 3*n^3 + 6*n^2 + n - 2)*x^3)*v[i - 1] + ((2*n + 1)*x + (4*n^3 - 4*n - 3)*x^2 + (2*n^5 + 3*n^4 - 4*n^3 - 3*n^2 + 2*n + 3)*x^3 - (2*n^5 + 3*n^4 - 3*n^2 + 1)*x^4)*deriv(v[i - 1], x))/(n^2 + n - 2 + (2*n^4 - 2*n^3 - 7*n^2 + 9*n - 2)*x + (n^6 - n^5 - 3*n^4 + 5*n^3 - 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 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

Proof chains ? JSON

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

• A352880 → reverse → (θ+1)²
• A108838 → (θ+1)²
• narayana → D → (θ+1)²

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