A038331 → x ↦ x+1 → D

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Submission text ?

Triangle begins:
  0:             170,              50;
  1:            4335,            2550,             375;
  2:           98260,           86700,           25500,            2500;
  3:         2088025,         2456500,         1083750,          212500,           15625;
  4:        42595710,        62640750,        36847500,        10837500,         1593750,          93750;
  5:       844814915,      1490849850,      1096213125,       429887500,        94828125,       11156250,         546875;
  6:     16413546920,     33792596600,     29816997000,     14616175000,      4298875000,      758625000,       74375000,       3125000;
  7:    313909084845,    738609611400,    760333423500,    447254955000,    164431968750,    38689875000,     5689687500,     478125000,     17578125;
  8:   5929393824850,  15695454242250,  18465240285000,  12672223725000,   5590686937500,  1644319687500,   322415625000,   40640625000,   2988281250,    97656250;
  9: 110879664524695, 326116660366750, 431624991661875, 338529405225000, 174243076218750, 61497556312500, 15072930468750, 2533265625000, 279404296875, 18261718750, 537109375;

The row polynomials satisfy (n+1)*P(n,x) = (17*n+34+(5*n+10)*x)*P(n-1,x) for n >= 1, with P(0,x) = 170 + 50*x.

P[0, x_] := 170 + 50*x;
P[n_, x_] := P[n, x] = Expand[Together[((17*n + 34 + (5*n + 10)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 170 + 50*x; for(n = 1, N, my(i = n + 1); v[i] = ((17*n + 34 + (5*n + 10)*x)*v[i - 1])/(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_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
        • 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 whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A038271 → basis A007318 → basis A007318
• A038331 → x ↦ x+1
• A038331 → x ↦ x+1 → reverse
• A013620 → basis A013611 → x ↦ x/(x+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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.