Hadamard-factor parent of A155495 → basis A013610

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Submission text ?

Triangle begins:
   1:       8,       24,         9;
   2:      32,      144,       162,        27;
   3:     128,      768,      1440,       864,         81;
   4:     512,     3840,     10080,     10800,       4050,        243;
   5:    2048,    18432,     62208,     96768,      68040,      17496,        729;
   6:    8192,    86016,    354816,    725760,     762048,     381024,      71442,       2187;
   7:   32768,   393216,   1916928,   4866048,    6842880,    5225472,    1959552,     279936,      6561;
   8:  131072,  1769472,   9953280,  30191616,   53374464,   55427328,   32332608,    9447840,   1062882,     19683;
   9:  524288,  7864320,  50135040, 176947200,  377395200,  498161664,  400308480,  184757760,  43302600,   3936600,    59049;
  10: 2097152, 34603008, 246546432, 992673792, 2481684480, 3985293312, 4109833728, 2642035968, 990763488, 190531440, 14289858, 177147;

The row polynomials satisfy P(n,x) = -9*x^2*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 3, with P(1,x) = 8 + 24*x + 9*x^2, P(2,x) = 32 + 144*x + 162*x^2 + 27*x^3.

P[1, x_] := 8 + 24*x + 9*x^2;
P[2, x_] := 32 + 144*x + 162*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[-9*x^2*P[n - 2, x] + (4 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 8 + 24*x + 9*x^2; v[2] = 32 + 144*x + 162*x^2 + 27*x^3; for(n = 3, N, my(i = n); v[i] = -9*x^2*v[i - 2] + (4 + 6*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
    • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
        • ↑ Three-term recurrence criterion (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

• A086645 → basis A013610 → reverse
• A086645 → basis A013610
• A034870 → Veronese 2 → basis A013610

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [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
3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023