Triangle of unsigned 2-Lah numbers.

Matches ?

Row sums ? A062147

Submission text ?

Triangle begins:
  0:          20,          10,           1;
  1:         120,          90,          18,           1;
  2:         840,         840,         252,          28,          1;
  3:        6720,        8400,        3360,         560,         40,         1;
  4:       60480,       90720,       45360,       10080,       1080,        54,         1;
  5:      604800,     1058400,      635040,      176400,      25200,      1890,        70,       1;
  6:     6652800,    13305600,     9313920,     3104640,     554400,     55440,      3080,      88,      1;
  7:    79833600,   179625600,   143700480,    55883520,   11975040,   1496880,    110880,    4752,    108,     1;
  8:  1037836800,  2594592000,  2335132800,  1037836800,  259459200,  38918880,   3603600,  205920,   7020,   130,   1;
  9: 14529715200, 39956716800, 39956716800, 19978358400, 5708102400, 998917920, 110990880, 7927920, 360360, 10010, 154, 1;

The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + (4+2*x)*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 20 + 10*x + x^2.

P[0, x_] := 20 + 10*x + x^2;
P[n_, x_] := P[n, x] = Expand[(4 + x)*P[n - 1, x] + (4 + 2*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 20 + 10*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (4 + x)*v[i - 1] + (4 + 2*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), 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, Laguerre’s theorem on differential operators, Iterated real-rootedness preservers
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Ray basis map theorem, Laguerre’s theorem on differential operators, Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem, Laguerre’s theorem on differential operators, Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
  by Iterated real-rootedness preservers
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
  by Row-hypergeometric rows (multiplier sequences)
• 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

• 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)
• Ore-algebra identity of two definitions polytool ? step › direct check
• 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

Used in 6

• one_plus_derivative of Triangle of unsigned 2-Lah numbers. Image under a real-rootedness preserver (derivative polynomial, shift 0)
• derivative of Triangle of unsigned 2-Lah numbers. Image under a real-rootedness preserver (derivative polynomial, shift 0)
• shift_x_plus_one of Triangle of unsigned 2-Lah numbers. Image under a real-rootedness preserver (affine, shift 0)
• reversal of Triangle of unsigned 2-Lah numbers. Image under a real-rootedness preserver (reversal, shift 0)
• mobius_x_over_x_plus_one of Triangle of unsigned 2-Lah numbers. Image under a real-rootedness preserver (mobius, shift 0)
• theta_plus_one of Triangle of unsigned 2-Lah numbers. Image under a real-rootedness preserver (euler polynomial, shift 0)

Related ?

Same fingerprint ?

• A136124 → basis A008277 → D+1
• A143491 → basis A008277 → D
• A143497 → reverse
• A136124 → basis A008277 → D

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.