A013612 → x ↦ x+1 → θ+1

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Submission text ?

Triangle begins:
  0:        36,        120,          75;
  1:       216,       1080,        1350,          500;
  2:      1296,       8640,       16200,        12000,         3125;
  3:      7776,      64800,      162000,       180000,        93750,        18750;
  4:     46656,     466560,     1458000,      2160000,      1687500,       675000,       109375;
  5:    279936,    3265920,    12247200,     22680000,     23625000,     14175000,      4593750,       625000;
  6:   1679616,   22394880,    97977600,    217728000,    283500000,    226800000,    110250000,     30000000,      3515625;
  7:  10077696,  151165440,   755827200,   1959552000,   3061800000,   3061800000,   1984500000,    810000000,    189843750,    19531250;
  8:  60466176, 1007769600,  5668704000,  16796160000,  30618000000,  36741600000,  29767500000,  16200000000,   5695312500,  1171875000,  107421875;
  9: 362797056, 6651279360, 41570496000, 138568320000, 288684000000, 404157600000, 392931000000, 267300000000, 125296875000, 38671875000, 7089843750, 585937500;

The row polynomials satisfy (6+(5*n+10)*x)*P(n,x) = (36+(30*n+120)*x+(25*n+75)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 120*x + 75*x^2.

P[0, x_] := 36 + 120*x + 75*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((36 + (30*n + 120)*x + (25*n + 75)*x^2)*P[n - 1, x])/(6 + (5*n + 10)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 36 + 120*x + 75*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((36 + (30*n + 120)*x + (25*n + 75)*x^2)*v[i - 1])/(6 + (5*n + 10)*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 Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+5x)^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 of coefficients in expansion of (1+5x)^n.?
        • ↑ 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 A013609 → θ+1
• A038336 → θ+1
• A038259 → θ+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.