Hadamard-factor parent of A155495 → basis A013611

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Submission text ?

Triangle begins:
   1:       8,       32,        16;
   2:      32,      192,       288,         64;
   3:     128,     1024,      2560,       2048,        256;
   4:     512,     5120,     17920,      25600,      12800,        1024;
   5:    2048,    24576,    110592,     229376,     215040,       73728,        4096;
   6:    8192,   114688,    630784,    1720320,    2408448,     1605632,      401408,       16384;
   7:   32768,   524288,   3407872,   11534336,   21626880,    22020096,    11010048,     2097152,      65536;
   8:  131072,  2359296,  17694720,   71565312,  168689664,   233570304,   181665792,    70778880,   10616832,     262144;
   9:  524288, 10485760,  89128960,  419430400, 1192755200,  2099249152,  2249195520,  1384120320,  432537600,   52428800,   1048576;
  10: 2097152, 46137344, 438304768, 2353004544, 7843348480, 16793993216, 23091740672, 19792920576, 9896460288, 2537553920, 253755392, 4194304;

The row polynomials satisfy P(n,x) = -16*x^2*P(n-2,x) + (4+8*x)*P(n-1,x) for n >= 3, with P(1,x) = 8 + 32*x + 16*x^2, P(2,x) = 32 + 192*x + 288*x^2 + 64*x^3.

P[1, x_] := 8 + 32*x + 16*x^2;
P[2, x_] := 32 + 192*x + 288*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[-16*x^2*P[n - 2, x] + (4 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 8 + 32*x + 16*x^2; v[2] = 32 + 192*x + 288*x^2 + 64*x^3; for(n = 3, N, my(i = n); v[i] = -16*x^2*v[i - 2] + (4 + 8*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 ?

• A084534 → reverse
• A034870 → Veronese 2 → basis A013611
• A084534: Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).
• A086645 → basis A013611 → reverse
• A086645 → basis A013611

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