A001263 → θ+1 → (θ+1)²

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Triangle begins:
  0: 1;
  1: 1,   8;
  2: 1,  24,    27;
  3: 1,  48,   162,     64;
  4: 1,  80,   540,    640,    125;
  5: 1, 120,  1350,   3200,   1875,     216;
  6: 1, 168,  2835,  11200,  13125,    4536,    343;
  7: 1, 224,  5292,  31360,  61250,   42336,   9604,    512;
  8: 1, 288,  9072,  75264, 220500,  254016, 115248,  18432,   729;
  9: 1, 360, 14580, 161280, 661500, 1143072, 864360, 276480, 32805, 1000;

The row polynomials satisfy (n^2+n-(4*n^4+6*n^3+n^2-n)*x+(n^6+3*n^5+3*n^4+n^3)*x^2)*P(n,x) = (n^2+n-(4*n^4+6*n^3-11*n^2-5*n)*x+(n^6+3*n^5-18*n^4-43*n^3-12*n^2-3*n)*x^2+(3*n^6+15*n^5+24*n^4+15*n^3+6*n^2+n)*x^3)*P(n-1,x) + ((2*n+1)*x-(8*n^3+20*n^2+12*n+3)*x^2+(2*n^5+11*n^4+28*n^3+35*n^2+16*n+3)*x^3-(2*n^5+11*n^4+20*n^3+15*n^2+6*n+1)*x^4)*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[((n^2 + n - (4*n^4 + 6*n^3 - 11*n^2 - 5*n)*x + (n^6 + 3*n^5 - 18*n^4 - 43*n^3 - 12*n^2 - 3*n)*x^2 + (3*n^6 + 15*n^5 + 24*n^4 + 15*n^3 + 6*n^2 + n)*x^3)*P[n - 1, x] + ((2*n + 1)*x - (8*n^3 + 20*n^2 + 12*n + 3)*x^2 + (2*n^5 + 11*n^4 + 28*n^3 + 35*n^2 + 16*n + 3)*x^3 - (2*n^5 + 11*n^4 + 20*n^3 + 15*n^2 + 6*n + 1)*x^4)*D[P[n - 1, x], x])/(n^2 + n - (4*n^4 + 6*n^3 + n^2 - n)*x + (n^6 + 3*n^5 + 3*n^4 + n^3)*x^2)]];
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] = ((n^2 + n - (4*n^4 + 6*n^3 - 11*n^2 - 5*n)*x + (n^6 + 3*n^5 - 18*n^4 - 43*n^3 - 12*n^2 - 3*n)*x^2 + (3*n^6 + 15*n^5 + 24*n^4 + 15*n^3 + 6*n^2 + n)*x^3)*v[i - 1] + ((2*n + 1)*x - (8*n^3 + 20*n^2 + 12*n + 3)*x^2 + (2*n^5 + 11*n^4 + 28*n^3 + 35*n^2 + 16*n + 3)*x^3 - (2*n^5 + 11*n^4 + 20*n^3 + 15*n^2 + 6*n + 1)*x^4)*deriv(v[i - 1], x))/(n^2 + n - (4*n^4 + 6*n^3 + n^2 - n)*x + (n^6 + 3*n^5 + 3*n^4 + n^3)*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
• 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 theta_plus_one 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 call…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 ?

• A367270 → (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