A143497 → D

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Submission text ?

Triangle begins:
  0:          10,           2;
  1:          90,          36,           3;
  2:         840,         504,          84,           4;
  3:        8400,        6720,        1680,         160,          5;
  4:       90720,       90720,       30240,        4320,        270,         6;
  5:     1058400,     1270080,      529200,      100800,       9450,       420,        7;
  6:    13305600,    18627840,     9313920,     2217600,     277200,     18480,      616,       8;
  7:   179625600,   287400960,   167650560,    47900160,    7484400,    665280,    33264,     864,     9;
  8:  2594592000,  4670265600,  3113510400,  1037836800,  194594400,  21621600,  1441440,   56160,  1170,   10;
  9: 39956716800, 79913433600, 59935075200, 22832409600, 4994589600, 665945280, 55495440, 2882880, 90090, 1540, 11;

The row polynomials satisfy (n+1)*P(n,x) = (n^2+7*n+10+(n+2)*x)*P(n-1,x) + (n+2)*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 10 + 2*x.

P[0, x_] := 10 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 7*n + 10 + (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] = 10 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 7*n + 10 + (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

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from Triangle of unsigned 2-Lah numbers.?
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from basis_map_A007318 of Unsigned 2-Stirling numbers of the first kind.?
        • Geometric basis map (elementary) polytool ? 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
        • Geometric basis map (elementary) polytool ? 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)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? 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)
    • 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 (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
        • 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)
    • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • 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

Related ?

Same fingerprint ?

• A143497 → D+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.