A038329 → basis A013610 → θ+1

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Submission text ?

Triangle begins:
  0:           225,             540,             243;
  1:          3375,           12150,           10935,             2916;
  2:         50625,          243000,          328050,           174960,            32805;
  3:        759375,         4556250,         8201250,          6561000,          2460375,           354294;
  4:      11390625,        82012500,       184528125,        196830000,        110716875,         31886460,          3720087;
  5:     170859375,      1435218750,      3875090625,       5166787500,       3875090625,       1674039150,        390609135,        38263752;
  6:    2562890625,     24603750000,     77501812500,     124002900000,     116252718750,      66961566000,      23436548100,      4591650240,       387420489;
  7:   38443359375,    415188281250,   1494677812500,    2790065250000,    3138823406250,    2259952852500,    1054644664500,    309936391200,     52301766015,     3874204890;
  8:  576650390625,   6919804687500,  28025208984375,   59787112500000,   78470585156250,   67798585575000,   39549174918750,  15496819560000,   3922632451125,   581130733500,   38354628411;
  9: 8649755859375, 114176777343750, 513795498046875, 1233109195312500, 1849663792968750, 1864461103312500, 1305122772318750, 639243806850000, 215744784811875, 47943285513750, 6328513687815, 376572715308;

The row polynomials satisfy (5+(3*n+6)*x)*P(n,x) = (75+(45*n+180)*x+(27*n+81)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 540*x + 243*x^2.

P[0, x_] := 225 + 540*x + 243*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((75 + (45*n + 180)*x + (27*n + 81)*x^2)*P[n - 1, x])/(5 + (3*n + 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 225 + 540*x + 243*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((75 + (45*n + 180)*x + (27*n + 81)*x^2)*v[i - 1])/(5 + (3*n + 6)*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_A013610 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

Related ?

Same fingerprint ?

• A013620 → basis A007318 → 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.