A038330 → basis A013609 → D+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:             512,             384,               64;
  1:           10240,           12288,             4608,              512;
  2:          196608,          327680,           196608,            49152,             4096;
  3:         3670016,         7864320,          6553600,          2621440,           491520,            32768;
  4:        67108864,       176160768,        188743680,        104857600,         31457280,          4718592,          262144;
  5:      1207959552,      3758096384,       4932501504,       3523215360,       1468006400,        352321536,        44040192,         2097152;
  6:     21474836480,     77309411328,     120259084288,     105226698752,      56371445760,      18790481920,      3758096384,       402653184,       16777216;
  7:    377957122048,   1546188226560,    2783138807808,    2886218022912,    1894080577536,     811748818944,    225485783040,     38654705664,     3623878656,     134217728;
  8:   6597069766656,  30236569763840,   61847529062400,   74217034874880,   57724360458240,   30305289240576,  10823317585920,   2576980377600,   386547056640,   32212254720,   1073741824;
  9: 114349209288704, 580542139465728, 1330409069608960, 1814194185830400, 1632774767247360, 1015948744065024, 444477575528448, 136064563937280, 28346784153600, 3779571220480, 283467841536, 8589934592;

The row polynomials satisfy (n+3+x)*P(n,x) = (16*n+64+(8*n+48)*x+8*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 512 + 384*x + 64*x^2.

P[0, x_] := 512 + 384*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 64 + (8*n + 48)*x + 8*x^2)*P[n - 1, x])/(n + 3 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 512 + 384*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 64 + (8*n + 48)*x + 8*x^2)*v[i - 1])/(n + 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 basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • 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
        • 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 ?

• basis-map coefficients of A113214, shifted → x ↦ x+1 → D+1
• A038208 → basis A007318 → D+1
• A093375 → x ↦ x+1 → reverse
• A196347 → x ↦ x+1 → D+1
• A120910 → D → D+1
• A093375 → D → basis A007318
• A356546 → x ↦ x+1 → D+1
• A085880 → x ↦ x+1 → D+1
• A178820 → basis A007318 → D+1
• A003506 → x ↦ x+1 → D+1
• A156992 → basis A007318 → D+1
• A093375 → x ↦ x+1
• A038221 → x ↦ x+1 → D+1
• A178822 → x ↦ x+1 → D+1
• A028326 → basis A007318 → D+1
• A120910 → D+1
• A027466 → basis A013611 → D+1
• A103406 → x ↦ x/(x+1)
• A038338 → x ↦ x+1 → D+1
• A094305 → x ↦ x+1 → D+1
• A085881 → basis A007318 → D+1
• A178821 → basis A007318 → D+1
• A038325 → basis A007318 → D+1
• A038207 → D+1
• A135065 → x ↦ x+1 → 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.