A038284 → θ+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:         64,          192,          108;
  1:        512,         2304,         2592,           864;
  2:       4096,        24576,        41472,         27648,          6480;
  3:      32768,       245760,       552960,        552960,        259200,         46656;
  4:     262144,      2359296,      6635520,       8847360,       6220800,       2239488,        326592;
  5:    2097152,     22020096,     74317824,     123863040,     116121600,      62705664,      18289152,       2239488;
  6:   16777216,    201326592,    792723456,    1585446912,    1857945600,    1337720832,     585252864,     143327232,      15116544;
  7:  134217728,   1811939328,   8153726976,   19025362944,   26754416640,   24078974976,   14046068736,    5159780352,    1088391168,    100776960;
  8: 1073741824,  16106127360,  81537269760,  217432719360,  356725555200,  385263599616,  280921374720,  137594142720,   43535646720,   8062156800,   665127936;
  9: 8589934592, 141733920768, 797253304320, 2391759912960, 4484549836800, 5650532794368, 4944216195072, 3027071139840, 1277045637120, 354734899200, 58531258368, 4353564672;

The row polynomials satisfy (4+(3*n+6)*x)*P(n,x) = (32+(24*n+96)*x+(18*n+54)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 192*x + 108*x^2.

P[0, x_] := 64 + 192*x + 108*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((32 + (24*n + 96)*x + (18*n + 54)*x^2)*P[n - 1, x])/(4 + (3*n + 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 64 + 192*x + 108*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((32 + (24*n + 96)*x + (18*n + 54)*x^2)*v[i - 1])/(4 + (3*n + 6)*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)*8^(i-j)*6^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

Related ?

Same fingerprint ?

• A013610 → basis A007318 → θ+1
• A038330 → basis A013610 → θ+1
• A038335 → 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.