A269944 → basis A013609 → D

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Triangle begins:
  0:           0;
  1:           2;
  2:           6,           8;
  3:          34,          64,          24;
  4:         360,         776,         432,          64;
  5:        6320,       14632,        9672,        2240,        160;
  6:      167720,      405712,      295368,       84032,       9760,       384;
  7:     6294040,    15682224,    12145752,     3896832,     571200,     37632,      896;
  8:   317972800,   809287360,   654334272,   227230272,   38302080,   3252480,   132608,   2048;
  9: 20831772000, 53875569600, 44959589760, 16514859072, 3057710880, 303336192, 16208640, 436224, 4608;

The row polynomials satisfy P(n,x) = -(n^4-8*n^3+26*n^2-40*n+25+(4*n^2-16*n+20)*x+4*x^2)*P(n-2,x) + (2*n^2-6*n+7+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (4*n^2 - 16*n + 20)*x + 4*x^2)*P[n - 2, x] + (2*n^2 - 6*n + 7 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (4*n^2 - 16*n + 20)*x + 4*x^2)*v[i - 2] + (2*n^2 - 6*n + 7 + 4*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_A013609 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k >…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k)…?
        • 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 read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k)…?
        • ↑ 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.