A132393 → basis A008277 → (θ+1)²

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Triangle begins:
  0:      1;
  1:      1,        4;
  2:      2,       16,        9;
  3:      6,       72,       81,       16;
  4:     24,      384,      648,      256,       25;
  5:    120,     2400,     5400,     3200,      625,       36;
  6:    720,    17280,    48600,    38400,    11250,     1296,      49;
  7:   5040,   141120,   476280,   470400,   183750,    31752,    2401,     64;
  8:  40320,  1290240,  5080320,  6021120,  2940000,   677376,   76832,   4096,   81;
  9: 362880, 13063680, 58786560, 81285120, 47628000, 13716864, 2074464, 165888, 6561, 100;

The row polynomials satisfy (1-(4*n-1)*x+n^2*x^2)*P(n,x) = (n-(4*n^2-n-4)*x+(n^3-7*n-5)*x^2+(n^2+2*n+1)*x^3)*P(n-1,x) + (x-(4*n+3)*x^2+(n^2+2*n+1)*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[((n - (4*n^2 - n - 4)*x + (n^3 - 7*n - 5)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x] + (x - (4*n + 3)*x^2 + (n^2 + 2*n + 1)*x^3)*D[P[n - 1, x], x])/(1 - (4*n - 1)*x + n^2*x^2)]];
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] = ((n - (4*n^2 - n - 4)*x + (n^3 - 7*n - 5)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1] + (x - (4*n + 3)*x^2 + (n^2 + 2*n + 1)*x^3)*deriv(v[i - 1], x))/(1 - (4*n - 1)*x + n^2*x^2)); 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 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) [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 (above)
            • ↑ Ray basis map theorem (above)
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.