A143491 → basis A008277 → D

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Submission text ?

Triangle begins:
  0:           8,           2;
  1:          60,          30,           3;
  2:         480,         360,          72,           4;
  3:        4200,        4200,        1260,         140,          5;
  4:       40320,       50400,       20160,        3360,        240,         6;
  5:      423360,      635040,      317520,       70560,       7560,       378,        7;
  6:     4838400,     8467200,     5080320,     1411200,     201600,     15120,      560,       8;
  7:    59875200,   119750400,    83825280,    27941760,    4989600,    498960,    27720,     792,     9;
  8:   798336000,  1796256000,  1437004800,   558835200,  119750400,  14968800,  1108800,   47520,  1080,   10;
  9: 11416204800, 28540512000, 25686460800, 11416204800, 2854051200, 428107680, 39639600, 2265120, 77220, 1430, 11;

The row polynomials satisfy (n+1)*P(n,x) = (n^2+6*n+8+(n+2)*x)*P(n-1,x) + (n+2)*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 8 + 2*x.

P[0, x_] := 8 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 6*n + 8 + (n + 2)*x)*P[n - 1, x] + (n + 2)*x*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] = 8 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 6*n + 8 + (n + 2)*x)*v[i - 1] + (n + 2)*x*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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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 → D
• A136124 → basis A008277 → D+1
• A143497 → reverse
• Triangle of unsigned 2-Lah numbers.

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.