A038330 → D+1

Matches ?

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Submission text ?

Triangle begins:
  0:           240,            128,             16;
  1:          3456,           2880,            768,             64;
  2:         48384,          55296,          23040,           4096,            256;
  3:        663552,         967680,         552960,         153600,          20480,          1024;
  4:       8957952,       15925248,       11612160,        4423680,         921600,         98304,          4096;
  5:     119439360,      250822656,      222953472,      108380160,       30965760,       5160960,        458752,        16384;
  6:    1576599552,     3822059520,     4013162496,     2378170368,      867041280,     198180864,      27525120,      2097152,       65536;
  7:   20639121408,    56757583872,    68797071360,    48157949952,    21403533312,    6242697216,    1189085184,    141557760,     9437184,     262144;
  8:  268308578304,   825564856320,  1135151677440,   917294284800,   481579499520,  171228266496,   41617981440,   6794772480,   707788800,   41943040,   1048576;
  9: 3467372396544, 11805577445376, 18162426839040, 16648891269120, 10090237132800, 4237899595776, 1255673954304, 261598740480, 37371248640, 3460300800, 184549376, 4194304;

The row polynomials satisfy (n+4+x)*P(n,x) = (12*n+60+(4*n+32)*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 240 + 128*x + 16*x^2.

P[0, x_] := 240 + 128*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 60 + (4*n + 32)*x + 4*x^2)*P[n - 1, x])/(n + 4 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 240 + 128*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 60 + (4*n + 32)*x + 4*x^2)*v[i - 1])/(n + 4 + 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)*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 ?

• A038243 → basis A013609 → D+1
• A027465 → D+1
• A027465 → D → D+1
• A093375 → x ↦ x+1 → x ↦ x+1
• A038279 → basis A013610 → D+1
• A120910 → 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.