A038328 → x ↦ x+1 → θ+1

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Submission text ?

Triangle begins:
  0:           196,            112,             12;
  1:          2744,           2352,            504,            32;
  2:         38416,          43904,          14112,          1792,            80;
  3:        537824,         768320,         329280,         62720,          5600,          192;
  4:       7529536,       12907776,        6914880,       1756160,        235200,        16128,          448;
  5:     105413504,      210827008,      135531648,      43025920,       7683200,       790272,        43904,        1024;
  6:    1475789056,     3373232128,     2529924096,     963780608,     215129600,     29503488,      2458624,      114688,       2304;
  7:   20661046784,    53128406016,    45538633728,   20239392768,    5421265920,    929359872,    103262208,     7225344,     290304,     5120;
  8:  289254654976,   826441871360,   796926090240,  404787855360,  126496204800,  26022076416,   3614177280,   337182720,   20321280,   716800,   11264;
  9: 4049565169664, 12727204818944, 13636290877440, 7792166215680, 2782916505600, 667899961344, 111316660224, 12981534720, 1043159040, 55193600, 1734656, 24576;

The row polynomials satisfy (7+(n+2)*x)*P(n,x) = (98+(14*n+56)*x+(2*n+6)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 112*x + 12*x^2.

P[0, x_] := 196 + 112*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((98 + (14*n + 56)*x + (2*n + 6)*x^2)*P[n - 1, x])/(7 + (n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 196 + 112*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((98 + (14*n + 56)*x + (2*n + 6)*x^2)*v[i - 1])/(7 + (n + 2)*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 whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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)*12^(i-j)*2^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 ?

• A027466 → 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.