A038284 → D+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:         160,          168,            36;
  1:        1664,         2880,          1512,           216;
  2:       16384,        39936,         34560,         12096,          1296;
  3:      155648,       491520,        599040,        345600,         90720,          7776;
  4:     1441792,      5603328,       8847360,       7188480,       3110400,        653184,         46656;
  5:    13107200,     60555264,     117669888,     123863040,      75479040,      26127360,       4572288,        279936;
  6:   117440512,    629145600,    1453326336,    1882718208,    1486356480,     724598784,     209018880,      31352832,      1679616;
  7:  1040187392,   6341787648,   16986931200,   26159874048,   25416695808,   16052649984,    6521389056,    1612431360,    211631616,    10077696;
  8:  9126805504,  62411243520,  190253629440,  339738624000,  392398110720,  305000349696,  160526499840,   55897620480,  12093235200,  1410877440,   60466176;
  9: 79456894976, 602369163264, 2059571036160, 4185579847680, 5605687296000, 5179655061504, 3355003846656, 1513535569920, 461155368960, 88683724800, 9311791104, 362797056;

The row polynomials satisfy (3*n+7+3*x)*P(n,x) = (24*n+80+(18*n+84)*x+18*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 160 + 168*x + 36*x^2.

P[0, x_] := 160 + 168*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((24*n + 80 + (18*n + 84)*x + 18*x^2)*P[n - 1, x])/(3*n + 7 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 160 + 168*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24*n + 80 + (18*n + 84)*x + 18*x^2)*v[i - 1])/(3*n + 7 + 3*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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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 ?

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