A103327 → basis A013609 → θ+1

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Triangle begins:
  0:      1;
  1:      4,       4;
  2:     16,      48,       12;
  3:     64,     320,      288,       32;
  4:    256,    1792,     2880,     1280,       80;
  5:   1024,    9216,    21504,    17920,     4800,      192;
  6:   4096,   45056,   138240,   172032,    89600,    16128,      448;
  7:  16384,  212992,   811008,  1351680,  1075200,   387072,    50176,    1024;
  8:  65536,  983040,  4472832,  9371648, 10137600,  5677056,  1505280,  147456,   2304;
  9: 262144, 4456448, 23592960, 59637760, 82001920, 63258624, 26492928, 5406720, 414720, 5120;

The row polynomials satisfy (4*n^2-2*n+(4*n^3-4*n^2+n)*x)*P(n,x) = (16*n^2-8*n+(16*n^3+32*n^2-36*n)*x+(24*n^3-26*n+6)*x^2)*P(n-1,x) - (8*n*x+(8*n^2+24*n-2)*x^2+(16*n^2+16*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[((16*n^2 - 8*n + (16*n^3 + 32*n^2 - 36*n)*x + (24*n^3 - 26*n + 6)*x^2)*P[n - 1, x] - (8*n*x + (8*n^2 + 24*n - 2)*x^2 + (16*n^2 + 16*n - 4)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 2*n + (4*n^3 - 4*n^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] = ((16*n^2 - 8*n + (16*n^3 + 32*n^2 - 36*n)*x + (24*n^3 - 26*n + 6)*x^2)*v[i - 1] - (8*n*x + (8*n^2 + 24*n - 2)*x^2 + (16*n^2 + 16*n - 4)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 2*n + (4*n^3 - 4*n^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 read by rows: T(n,k) = binomial(2n+1, 2k+1).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
                • ↑ Liu–Wang theorem for three-term recurrences (above)
                • ↑ Three-term recurrence criterion (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Three-term recurrence criterion (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A123519 → reverse → 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