A038321 → basis A007318 → x ↦ x/(x+1)

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Submission text ?

Triangle begins:
  0:            324,             900,              625;
  1:           5832,           24300,            33750,             15625;
  2:         104976,          583200,          1215000,           1125000,            390625;
  3:        1889568,        13122000,         36450000,          50625000,          35156250,            9765625;
  4:       34012224,       283435200,        984150000,        1822500000,        1898437500,         1054687500,          244140625;
  5:      612220032,      5952139200,      24800580000,       57408750000,       79734375000,        66445312500,        30761718750,         6103515625;
  6:    11019960576,    122444006400,     595213920000,     1653372000000,     2870437500000,      3189375000000,      2214843750000,       878906250000,       152587890625;
  7:   198359290368,   2479491129600,   13774950720000,    44641044000000,    93002175000000,    129169687500000,    119601562500000,     71191406250000,     24719238281250,     3814697265625;
  8:  3570467226624,  49589822592000,  309936391200000,  1147912560000000,  2790065250000000,   4650108750000000,   5382070312500000,   4271484375000000,   2224731445312500,   686645507812500,    95367431640625;
  9: 64268410079232, 981878487321600, 6818600606400000, 28410835860000000, 78918988500000000, 153453588750000000, 213129984375000000, 211438476562500000, 146832275390625000, 67977905273437500, 18882751464843750, 2384185791015625;

The row polynomials satisfy (1+x)*P(n,x) = (18+43*x+25*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 900*x + 625*x^2.

P[0, x_] := 324 + 900*x + 625*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((18 + 43*x + 25*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 324 + 900*x + 625*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((18 + 43*x + 25*x^2)*v[i - 1])/(1 + 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 [-1, 0]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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
        • 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)*11^(i-j)*7^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A038277 → x ↦ x/(x+1) → 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.