A132393 → basis A013610 → (θ+1)²

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Triangle begins:
  0:      1;
  1:      1,       12;
  2:      2,       36,       81;
  3:      6,      132,      486,       432;
  4:     24,      600,     2835,      4320,      2025;
  5:    120,     3288,    18225,     36720,     30375,      8748;
  6:    720,    21168,   131544,    317520,    354375,    183708,     35721;
  7:   5040,   156816,  1063692,   2924208,   3969000,   2816856,   1000188,    139968;
  8:  40320,  1315008,  9568044,  29066688,  45459225,  39680928,  19503666,   5038848,   531441;
  9: 362880, 12318912, 94988700, 312629760, 545383125, 553512204, 337563450, 121772160, 23914845, 1968300;

The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(12*n-12)*x-81*x^2)*P(n-2,x) + ((12*n-12)*x^2+36*x^3)*d/dx P(n-2,x) + (2*n-1+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x.

P[0, x_] := 1;
P[1, x_] := 1 + 12*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + 1 + (12*n - 12)*x - 81*x^2)*P[n - 2, x] + ((12*n - 12)*x^2 + 36*x^3)*D[P[n - 2, x], x] + (2*n - 1 + 12*x)*P[n - 1, x] - 12*x^2*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] = 1 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n + 1 + (12*n - 12)*x - 81*x^2)*v[i - 2] + ((12*n - 12)*x^2 + 36*x^3)*deriv(v[i - 2], x) + (2*n - 1 + 12*x)*v[i - 1] - 12*x^2*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_A013610 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.