A035607 → basis A007318 → basis A013609

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Submission text ?

Triangle begins:
  0:     1;
  1:     1;
  2:     5,      4;
  3:    17,     24,      8;
  4:    61,    124,     80,     16;
  5:   217,    576,    552,    224,     32;
  6:   773,   2532,   3216,   1968,    576,     64;
  7:  2753,  10728,  16968,  13888,   6176,   1408,   128;
  8:  9805,  44300,  83856,  85968,  51392,  17856,  3328,  256;
  9: 34921, 179472, 395560, 487328, 366240, 171520, 48768, 7680, 512;

The row polynomials satisfy P(n,x) = (2+2*x)*P(n-2,x) + (3+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] + (3 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] + (3 + 2*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 - 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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A007318 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercu…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
        • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…?
        • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

• A249252 → reverse → x ↦ x/(x+1)
• A249252 → basis A007318 → reverse
• A249252 → basis A007318

References

1. [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