A132393 → basis A013611 → D+1

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Submission text ?

Triangle begins:
  0:       1;
  1:       5,        4;
  2:      14,       44,        16;
  3:      50,      236,       288,        64;
  4:     224,     1320,      2480,      1664,       256;
  5:    1216,     8296,     19920,     20800,      8960,      1024;
  6:    7776,    59024,    167104,    226240,    152320,     46080,      4096;
  7:   57312,   472496,   1509760,   2440256,   2150400,   1017856,    229376,    16384;
  8:  478656,  4218304,  14808512,  27293952,  28971264,  18063360,   6365184,  1114112,   65536;
  9: 4469184, 41632704, 157709760, 322104320, 392904960, 297034752, 138485760, 37847040, 5308416, 262144;

The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(8*n-8)*x+16*x^2)*P(n-2,x) + (2*n-1+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 4*x.

P[0, x_] := 1;
P[1, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + 1 + (8*n - 8)*x + 16*x^2)*P[n - 2, x] + (2*n - 1 + 8*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] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n + 1 + (8*n - 8)*x + 16*x^2)*v[i - 2] + (2*n - 1 + 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 - 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013611 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 …?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.?
        • Parent-row provers of polytool polytool ? step › direct check
        • 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.