A143491 → basis A008277 → (θ+1)²

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Submission text ?

Triangle begins:
  0:         12,          32,            9;
  1:         60,         240,          135,           16;
  2:        360,        1920,         1620,          384,          25;
  3:       2520,       16800,        18900,         6720,         875,          36;
  4:      20160,      161280,       226800,       107520,       21000,        1728,         49;
  5:     181440,     1693440,      2857680,      1693440,      441000,       54432,       3087,        64;
  6:    1814400,    19353600,     38102400,     27095040,     8820000,     1451520,     123480,      5120,       81;
  7:   19958400,   239500800,    538876800,    447068160,   174636000,    35925120,    4074840,    253440,     8019,    100;
  8:  239500800,  3193344000,   8083152000,   7664025600,  3492720000,   862202880,  122245200,  10137600,   481140,  12000,   121;
  9: 3113510400, 45664819200, 128432304000, 136994457600, 71351280000, 20549168640, 3496212720, 362419200, 22934340, 858000, 17303, 144;

The row polynomials satisfy (1+(2*n+3)*x+(n^2+4*n+4)*x^2)*P(n,x) = (n+4+(2*n^2+11*n+16)*x+(n^3+8*n^2+23*n+23)*x^2+(n^2+6*n+9)*x^3)*P(n-1,x) + (x+(2*n+5)*x^2+(n^2+6*n+9)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 12 + 32*x + 9*x^2.

P[0, x_] := 12 + 32*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (2*n^2 + 11*n + 16)*x + (n^3 + 8*n^2 + 23*n + 23)*x^2 + (n^2 + 6*n + 9)*x^3)*P[n - 1, x] + (x + (2*n + 5)*x^2 + (n^2 + 6*n + 9)*x^3)*D[P[n - 1, x], x])/(1 + (2*n + 3)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 12 + 32*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (2*n^2 + 11*n + 16)*x + (n^3 + 8*n^2 + 23*n + 23)*x^2 + (n^2 + 6*n + 9)*x^3)*v[i - 1] + (x + (2*n + 5)*x^2 + (n^2 + 6*n + 9)*x^3)*deriv(v[i - 1], x))/(1 + (2*n + 3)*x + (n^2 + 4*n + 4)*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 Unsigned 2-Stirling numbers of the first kind.?
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from Unsigned 2-Stirling numbers of the first kind.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A136124 → basis A008277 → (θ+1)²

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.