A085881 → basis A007318 → basis A008277

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Triangle begins:
  0:            1;
  1:            3,             1;
  2:           27,            21,              3;
  3:          405,           555,            180,             15;
  4:         8505,         18375,          10185,           1890,            105;
  5:       229635,        738045,         623700,         193725,          23625,           945;
  6:      7577955,      34999965,       42421995,       19646550,        3950100,        343035,        10395;
  7:    295540245,    1918511595,     3212429220,     2083916835,      614864250,      87026940,      5675670,      135135;
  8:  13299311025,  119543799375,   269709865425,   235738953450,    95819498775,   19666196550,   2071619550,   105405300,    2027025;
  9: 678264862275, 8355066644925, 24957583150500, 28630440964125, 15410082562875, 4303534209975, 651283132500, 53136433350, 2170943775, 34459425;

The row polynomials satisfy P(n,x) = (6*n-3+(2*n-1)*x)*P(n-1,x) + (2*n-1)*x*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[(6*n - 3 + (2*n - 1)*x)*P[n - 1, x] + (2*n - 1)*x*D[P[n - 1, x], 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] = (6*n - 3 + (2*n - 1)*x)*v[i - 1] + (2*n - 1)*x*deriv(v[i - 1], 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 Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem

Proof chains ? JSON

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

  • from basis_map_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A135065 → x ↦ x+1 → basis A008277
• A038207 → basis A008277 → reverse
• basis-map coefficients of A113214, shifted → x ↦ x+1 → basis A008277
• A120910 → D → basis A008277
• A143495 → reverse
• A356546 → x ↦ x+1 → basis A008277
• A003506 → x ↦ x+1 → basis A008277
• A038325 → basis A007318 → basis A008277
• A178822 → x ↦ x+1 → basis A008277
• A027466 → basis A013611 → basis A008277
• A196347 → x ↦ x+1 → basis A008277
• A028326 → basis A007318 → basis A008277
• A038330 → basis A013609 → basis A008277
• A120910 → basis A008277
• A094305 → x ↦ x+1 → basis A008277
• A085880 → x ↦ x+1 → basis A008277
• A143495: Triangle read by rows: 3-Stirling numbers of the second kind.
• A038207 → basis A008277
• A178821 → basis A007318 → basis A008277
• A038221 → x ↦ x+1 → basis A008277
• A178820 → basis A007318 → basis A008277
• A038208 → basis A007318 → basis A008277
• A156992 → basis A007318 → basis A008277
• A038338 → x ↦ x+1 → basis A008277

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.