A090582 → D

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Triangle begins:
  0:        0;
  1:        1;
  2:        6,        2;
  3:       36,       28,        3;
  4:      240,      300,       90,        4;
  5:     1800,     3120,     1620,      248,        5;
  6:    15120,    33600,    25200,     7224,      630,       6;
  7:   141120,   383040,   378000,   163296,    28980,    1524,      7;
  8:  1451520,  4656960,  5715360,  3336480,   932400,  108900,   3570,    8;
  9: 16329600, 60480000, 88905600, 65741760, 25515000, 4911120, 391860, 8176, 9;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - n^2 + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] + (n^2*x + (2*n^2 - n)*x^2 + (n^2 - n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - n + (2*n^2 - 4*n + 2)*x)*v[i - 1] - ((n - 1)*x + (n - 1)*x^2)*deriv(v[i - 1], x))/(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 T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= …?
    • Parent-row provers of polytool polytool ? step › direct check
    • 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum o…?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum o…?
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A257609 → D (shift +2) → x ↦ x+1
• A008292 → D → x ↦ x+1
• A173018 → x ↦ x+1 → D

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.