A090582 → θ+1

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Submission text ?

Triangle begins:
  0:       1;
  1:       2,        2;
  2:       6,       12,        3;
  3:      24,       72,       42,         4;
  4:     120,      480,      450,       120,        5;
  5:     720,     3600,     4680,      2160,      310,        6;
  6:    5040,    30240,    50400,     33600,     9030,      756,       7;
  7:   40320,   282240,   574560,    504000,   204120,    34776,    1778,      8;
  8:  362880,  2903040,  6985440,   7620480,  4170600,  1118880,  127050,   4080,    9;
  9: 3628800, 32659200, 90720000, 118540800, 82177200, 30618000, 5729640, 447840, 9198, 10;

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

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

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

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 ?

• A173018 → x ↦ x+1 → θ+1
• A008292 → x ↦ x+1 → θ+1

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.