A088996 → D+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:     1;
  1:     1,      1;
  2:     1,      5,       2;
  3:     2,     16,      25,        6;
  4:     6,     64,     167,      142,       24;
  5:    24,    316,    1133,     1633,      926,      120;
  6:   120,   1868,    8437,    16937,    16384,     6876,      720;
  7:   720,  12888,   69816,   178328,   242091,   173836,    57492,     5040;
  8:  5040, 101736,  640488,  1976776,  3435429,  3478092,  1970828,   535536,   40320;
  9: 40320, 904608, 6474660, 23355020, 49278057, 64126641, 51389416, 23938524, 5505264, 362880;

The row polynomials satisfy (n-1)*P(n,x) = -(n^3-4*n^2+4*n+(2*n^3-6*n^2+4*n)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) + (2*n^2-4*n+1+(2*n^2-2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n^2 + 4*n + (2*n^3 - 6*n^2 + 4*n)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] + (2*n^2 - 4*n + 1 + (2*n^2 - 2*n)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 4*n^2 + 4*n + (2*n^3 - 6*n^2 + 4*n)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] + (2*n^2 - 4*n + 1 + (2*n^2 - 2*n)*x)*v[i - 1])/(n - 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

Proof chains ? JSON

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

  • from Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.?
    • 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
    • Affine, reversal and Möbius substitutions (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
    • Affine, reversal and Möbius substitutions (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.