A038321 → θ+1

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Submission text ?

Triangle begins:
  0:          121,           308,            147;
  1:         1331,          5082,           4851,           1372;
  2:        14641,         74536,         106722,          60368,          12005;
  3:       161051,       1024870,        1956570,        1660120,         660275,         100842;
  4:      1771561,      13528284,       32283405,       36522640,       21789075,        6655572,         823543;
  5:     19487171,     173612978,      497164437,      703060820,      559252925,      256239522,       63412811,        6588344;
  6:    214358881,    2182563152,     7291745076,    12373870432,    12303564350,     7516359312,     2790163684,      579774272,       51883209;
  7:   2357947691,   27009219006,   103126108932,   204168862128,   243610574130,   186029892972,    92075401572,    28698826464,     5136437691,     403536070;
  8:  25937424601,  330112676740,  1417983997815,  3208367833440,  4466193859050,  4092657645384,  2532073543230,  1052290303680,   282504073005,   44388967700,   3107227739;
  9: 285311670611, 3994363388554, 19064007081735, 48526563480780, 77201350992150, 82535262515244, 61276179746166, 31831781686320, 11394330944535, 2685532545850, 375974556419, 23727920916;

The row polynomials satisfy (11+(7*n+14)*x)*P(n,x) = (121+(77*n+308)*x+(49*n+147)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 308*x + 147*x^2.

P[0, x_] := 121 + 308*x + 147*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((121 + (77*n + 308)*x + (49*n + 147)*x^2)*P[n - 1, x])/(11 + (7*n + 14)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 121 + 308*x + 147*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((121 + (77*n + 308)*x + (49*n + 147)*x^2)*v[i - 1])/(11 + (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 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

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.