A144084 → θ+1

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Submission text ?

Triangle begins:
  0: 1;
  1: 1,   2;
  2: 1,   8,    6;
  3: 1,  18,   54,     24;
  4: 1,  32,  216,    384,     120;
  5: 1,  50,  600,   2400,    3000,      720;
  6: 1,  72, 1350,   9600,   27000,    25920,     5040;
  7: 1,  98, 2646,  29400,  147000,   317520,   246960,    40320;
  8: 1, 128, 4704,  75264,  588000,  2257920,  3951360,  2580480,   362880;
  9: 1, 162, 7776, 169344, 1905120, 11430720, 35562240, 52254720, 29393280, 3628800;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (n^2 + 4*n - 2)*x + (2*n^3 + 2*n^2 - 2*n)*x^2)*v[i - 1] - (x^2 + (n^2 + 2*n)*x^3)*deriv(v[i - 1], x))/(1 + 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 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 T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.?
    • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A008277 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 …?
        • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

          • from Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.?
            • 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
            • Parent-row provers of polytool polytool ? step › direct check
        • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

          • from Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A395972 → 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.