A130123 → basis A008277 → θ+1

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Triangle begins:
  0:   1;
  1:   2,      4;
  2:   4,     24,       12;
  3:   8,    112,      144,       32;
  4:  16,    480,     1200,      640,        80;
  5:  32,   1984,     8640,     8320,      2400,      192;
  6:  64,   8064,    57792,    89600,     44800,     8064,      448;
  7: 128,  32512,   370944,   870912,    672000,   204288,    25088,    1024;
  8: 256, 130560,  2323200,  7956480,   8897280,  4064256,   827904,   73728,   2304;
  9: 512, 523264, 14330880, 69847040, 108864000, 70124544, 21073920, 3072000, 207360, 5120;

The row polynomials satisfy P(n,x) = -4*x^2*P(n-2,x) - 4*x^2*d/dx P(n-2,x) + (2+4*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[-4*x^2*P[n - 2, x] - 4*x^2*D[P[n - 2, x], x] + (2 + 4*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -4*x^2*v[i - 2] - 4*x^2*deriv(v[i - 2], x) + (2 + 4*x)*v[i - 1] + 2*x*deriv(v[i - 1], 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_A008277 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,…?
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …?
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A008277 → θ+1 → reverse
• basis-map coefficients of A021012 → basis A008277 → θ+1
• A321331: Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).
• basis-map coefficients of A112227 → basis A008277 → θ+1
• A048993 → D+1 → θ+1
• A008277 → θ+1

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.