A038329 → basis A013609 → θ+1

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Submission text ?

Triangle begins:
  0:           225,            360,             108;
  1:          3375,           8100,            4860,             864;
  2:         50625,         162000,          145800,           51840,            6480;
  3:        759375,        3037500,         3645000,         1944000,          486000,           46656;
  4:      11390625,       54675000,        82012500,        58320000,        21870000,         4199040,          326592;
  5:     170859375,      956812500,      1722262500,      1530900000,       765450000,       220449600,        34292160,        2239488;
  6:    2562890625,    16402500000,     34445250000,     36741600000,     22963500000,      8817984000,      2057529600,      268738560,      15116544;
  7:   38443359375,   276792187500,    664301250000,    826686000000,    620014500000,    297606960000,     92588832000,    18139852800,    2040733440,     100776960;
  8:  576650390625,  4613203125000,  12455648437500,  17714700000000,  15500362500000,   8928208800000,   3472081200000,   906992640000,  153055008000,   15116544000,    665127936;
  9: 8649755859375, 76117851562500, 228353554687500, 365365687500000, 365365687500000, 245525742000000, 114578679600000, 37413446400000, 8418025440000, 1247114880000, 109746109440, 4353564672;

The row polynomials satisfy (5+(2*n+4)*x)*P(n,x) = (75+(30*n+120)*x+(12*n+36)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 360*x + 108*x^2.

P[0, x_] := 225 + 360*x + 108*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((75 + (30*n + 120)*x + (12*n + 36)*x^2)*P[n - 1, x])/(5 + (2*n + 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 225 + 360*x + 108*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((75 + (30*n + 120)*x + (12*n + 36)*x^2)*v[i - 1])/(5 + (2*n + 4)*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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013610 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 basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
            • Geometric basis map (elementary) polytool ? step ›

              • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
                • 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
                • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
                • Parent-row provers of polytool polytool ? step › direct check
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038291 → basis A013611 → θ+1
• A038220 → basis A007318 → θ+1
• A038334 → x ↦ x+1 → 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.