A257620 → θ+1

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Submission text ?

Triangle begins:
  0:     1;
  1:     3,        6;
  2:     9,       72,         27;
  3:    27,      594,        891,         108;
  4:    81,     4212,      16038,        8424,          405;
  5:   243,    27702,     220158,      293544,        69255,         1458;
  6:   729,   174960,    2604717,     7045056,      4341195,       524880,        5103;
  7:  2187,  1080378,   28166373,   136635012,    170793765,     56332746,     3781323,      17496;
  8:  6561,  6587244,  287529264,  2315613096,   5123812950,   3473419644,   670901616,   26348976,     59049;
  9: 19683, 39877758, 2824904160, 35838176544, 128958488910, 154750186692, 62716808952, 7533077760, 179449911, 196830;

The row polynomials satisfy P(n,x) = -(9*n^2-9)*x^2*P(n-2,x) - ((9*n+9)*x^2-(9*n+9)*x^3)*d/dx P(n-2,x) + (3+6*n*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 9)*x^2*v[i - 2] - ((9*n + 9)*x^2 - (9*n + 9)*x^3)*deriv(v[i - 2], x) + (3 + 6*n*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], 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 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 read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    • 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 Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • Parent-row provers of polytool polytool ? step › direct check
    • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A008292 → θ+1 (shift +2) → reverse
• A334218 → D
• A257609 → θ+1
• A173018 → θ+1
• row-factor cofactor of A101842 → θ+1
• A008292 → θ+1
• A008292 → θ+1 (shift +2)
• A173018 → reverse (shift +1) → reverse → 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.