A137320 → basis A013610 → D+1

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Triangle begins:
  0:        1;
  1:        8,         6;
  2:       36,       102,         36;
  3:      180,       804,        972,        216;
  4:     1044,      6036,      11628,       8208,       1296;
  5:     6984,     48024,     121320,     137160,      64800,       7776;
  6:    53208,    415656,    1262664,    1888920,    1432080,     489888,      46656;
  7:   455904,   3933504,   13697712,   24845184,   25084080,   13771296,    3592512,     279936;
  8:  4345056,  40630464,  157434192,  329670432,  406126224,  298287360,  124758144,   25754112,   1679616;
  9: 45627840, 456268032, 1928069568, 4519529568, 6464235456, 5821992288, 3268532736, 1079993088, 181398528, 10077696;

The row polynomials satisfy P(n,x) = -(n^2+12*n*x+36*x^2)*P(n-2,x) + (2*n+1+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 6*x.

P[0, x_] := 1;
P[1, x_] := 8 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 12*n*x + 36*x^2)*P[n - 2, x] + (2*n + 1 + 12*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] = 8 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 12*n*x + 36*x^2)*v[i - 2] + (2*n + 1 + 12*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_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.