A038333 → x ↦ x+1 → θ+1

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Triangle begins:
  0:             361,             532,              147;
  1:            6859,           15162,             8379,             1372;
  2:          130321,          384104,           318402,           104272,            12005;
  3:         2476099,         9122470,         10082730,          4952920,          1140475,           100842;
  4:        47045881,       207992316,        287357805,        188210960,         65007075,         11495988,          823543;
  5:       893871739,      4610496338,       7643717613,       6258014420,       2881980325,        764483202,       109531219,         6588344;
  6:     16983563041,    100113634768,     193640846196,     190243638368,     109515252350,      38733815568,      8324372644,      1001428288,       51883209;
  7:    322687697779,   2139928943166,    4730369242788,    5421943693488,    3745421630370,    1655870615532,    474489240708,     85622118624,     8872028739,     403536070;
  8:   6131066257801,  45176277689060,  112346269516215,  147167043108960,  118605018295050,   62923083390216,  22538238933630,   5422734179520,   842842730205,   76671853300,   3107227739;
  9: 116490258898219, 944184203701354, 2608930036543215, 3844739001221580, 3541206974809350, 2191820738092524, 942098387425734, 283337860879920, 58718043537615, 8012208669850, 649410597451, 23727920916;

The row polynomials satisfy (19+(7*n+14)*x)*P(n,x) = (361+(133*n+532)*x+(49*n+147)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 532*x + 147*x^2.

P[0, x_] := 361 + 532*x + 147*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((361 + (133*n + 532)*x + (49*n + 147)*x^2)*P[n - 1, x])/(19 + (7*n + 14)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 361 + 532*x + 147*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((361 + (133*n + 532)*x + (49*n + 147)*x^2)*v[i - 1])/(19 + (7*n + 14)*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)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

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

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.