A139548 → x ↦ x+1 → θ+1

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Submission text ?

Triangle begins:
  0:         1;
  1:         8,         16,           6;
  2:        64,        256,         288,         128,           20;
  3:       512,       3072,        5760,        5120,         2400,          576,           56;
  4:      4096,      32768,       86016,      114688,        89600,        43008,        12544,         2048,          144;
  5:     32768,     327680,     1105920,     1966080,      2150400,      1548288,       752640,       245760,        51840,         6400,         352;
  6:    262144,    3145728,    12976128,    28835840,     40550400,     38928384,     26492928,     12976128,      4561920,      1126400,      185856,       18432,        832;
  7:   2097152,   29360128,   143130624,   381681664,    656015360,    787218432,    688816128,    449839104,    221405184,     82001920,    22550528,     4472832,     605696,      50176,      1920;
  8:  16777216,  268435456,  1509949440,  4697620480,   9542041600,  13740539904,  14694744064,  11995709440,   7591034880,   3748659200,  1443233792,   429391872,   96911360,   16056320,   1843200,   131072,    4352;
  9: 134217728, 2415919104, 15401484288, 54760833024, 128345702400, 215620780032, 272520708096, 266959060992, 206476148736, 127454412800, 63089934336, 25027411968, 7907966976, 1965293568, 376012800, 53477376, 5326848, 331776, 9728;

The row polynomials satisfy (2+(2*n-1)*x)*P(n,x) = (16+(16*n+24)*x+(16*n+12)*x^2+(4*n+2)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16 + (16*n + 24)*x + (16*n + 12)*x^2 + (4*n + 2)*x^3)*P[n - 1, x])/(2 + (2*n - 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16 + (16*n + 24)*x + (16*n + 12)*x^2 + (4*n + 2)*x^3)*v[i - 1])/(2 + (2*n - 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 (−∞, 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

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 T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
            • 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
            • 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 (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

• A034870 → 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.