A163936 → θ+1

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Submission text ?

Triangle begins:
  0:      1;
  1:      1;
  2:      2,       2;
  3:      6,      16,        3;
  4:     24,     116,       66,        4;
  5:    120,     888,      984,      208,        5;
  6:    720,    7416,    13200,     5808,      570,       6;
  7:   5040,   67968,   174420,   128480,    28050,    1440,      7;
  8:  40320,  682272,  2355912,  2576080,   979000,  119700,   3458,    8;
  9: 362880, 7467840, 33078888, 49760256, 28827500, 6375000, 470820, 8032, 9;

The row polynomials satisfy (n-1)*P(n,x) = -(n^3-3*n^2+2*n+(2*n^3-8*n^2+8*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) - ((n^2-2*n)*x+n*x^2-(n^2-n)*x^3)*d/dx P(n-2,x) + (2*n^2-3*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) = 1, P(1,x) = 1.

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 3*n^2 + 2*n + (2*n^3 - 8*n^2 + 8*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] - ((n^2 - 2*n)*x + n*x^2 - (n^2 - n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 3*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] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 3*n^2 + 2*n + (2*n^3 - 8*n^2 + 8*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] - ((n^2 - 2*n)*x + n*x^2 - (n^2 - n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 3*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 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
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by 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 Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).?
    • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).?
            • ↑ Liu–Wang theorem (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).?
            • ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

• A201637 → reverse (shift +1) → D
• A112007 → θ+1

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.