A137320 → basis A013610 → (θ+1)²

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Triangle begins:
  0:       1;
  1:       2,        24;
  2:       6,       120,        324;
  3:      24,       624,       2916,       3456;
  4:     120,      3696,      23004,      48384,       32400;
  5:     720,     25056,     187920,     535680,      648000,      279936;
  6:    5040,    192672,    1653696,    5754240,     9558000,     7558272,     2286144;
  7:   40320,   1662336,   15830640,   63673344,   130410000,   143047296,    80015040,    17915904;
  8:  362880,  15928704,  164917296,  741920256,  1770627600,  2414168064,  1888354944,   788299776,  136048896;
  9: 3628800, 167996160, 1864210464, 9178320384, 24661778400, 39439903104, 38599255296, 22681534464, 7346640384, 1007769600;

The row polynomials satisfy P(n,x) = -(n^2+24*n*x-324*x^2)*P(n-2,x) + (24*n*x^2+144*x^3)*d/dx P(n-2,x) + (2*n+1+24*x)*P(n-1,x) - 24*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 24*x.

P[0, x_] := 1;
P[1, x_] := 2 + 24*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 24*n*x - 324*x^2)*P[n - 2, x] + (24*n*x^2 + 144*x^3)*D[P[n - 2, x], x] + (2*n + 1 + 24*x)*P[n - 1, x] - 24*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] = 2 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 24*n*x - 324*x^2)*v[i - 2] + (24*n*x^2 + 144*x^3)*deriv(v[i - 2], x) + (2*n + 1 + 24*x)*v[i - 1] - 24*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 Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…?
        • 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 Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…?
        • ↑ 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.