A090582 → D+1

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Submission text ?

Triangle begins:
  0:        1;
  1:        3,        1;
  2:       12,        8,         1;
  3:       60,       64,        17,        1;
  4:      360,      540,       240,       34,        1;
  5:     2520,     4920,      3180,      788,       67,        1;
  6:    20160,    48720,     42000,    15624,     2436,      132,       1;
  7:   181440,   524160,    569520,   289296,    69804,     7320,     261,     1;
  8:  1814400,  6108480,   8043840,  5241600,  1766520,   295380,   21720,   518,    1;
  9: 19958400, 76809600, 119145600, 95376960, 41950440, 10014120, 1210380, 64156, 1031, 1;

The row polynomials satisfy (n+2*x)*P(n,x) = -(n^3-n+(2*n^3+n^2-5*n)*x+(n^3+4*n^2-6*n+1)*x^2+(3*n^2-2*n+1)*x^3+(2*n-2)*x^4)*P(n-2,x) + ((n^2+n)*x+(2*n^2+4*n)*x^2+(n^2+6*n+1)*x^3+(3*n+3)*x^4+2*x^5)*d/dx P(n-2,x) + (2*n^2+n+(2*n^2+3*n)*x+(5*n-3)*x^2+2*x^3)*P(n-1,x) - (n*x+(n+2)*x^2+2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n + (2*n^3 + n^2 - 5*n)*x + (n^3 + 4*n^2 - 6*n + 1)*x^2 + (3*n^2 - 2*n + 1)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] + ((n^2 + n)*x + (2*n^2 + 4*n)*x^2 + (n^2 + 6*n + 1)*x^3 + (3*n + 3)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n^2 + n + (2*n^2 + 3*n)*x + (5*n - 3)*x^2 + 2*x^3)*P[n - 1, x] - (n*x + (n + 2)*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - n + (2*n^3 + n^2 - 5*n)*x + (n^3 + 4*n^2 - 6*n + 1)*x^2 + (3*n^2 - 2*n + 1)*x^3 + (2*n - 2)*x^4)*v[i - 2] + ((n^2 + n)*x + (2*n^2 + 4*n)*x^2 + (n^2 + 6*n + 1)*x^3 + (3*n + 3)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (2*n^2 + n + (2*n^2 + 3*n)*x + (5*n - 3)*x^2 + 2*x^3)*v[i - 1] - (n*x + (n + 2)*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 2*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 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 ?

• A173018 → x ↦ x+1 → D+1
• A008292 → x ↦ x+1 → D+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.