A038328 → D

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Submission text ?

Triangle begins:
  0:            48,             8;
  1:           864,           288,            24;
  2:         13824,          6912,          1152,           64;
  3:        207360,        138240,         34560,         3840,          160;
  4:       2985984,       2488320,        829440,       138240,        11520,         384;
  5:      41803776,      41803776,      17418240,      3870720,       483840,       32256,        896;
  6:     573308928,     668860416,     334430208,     92897280,     15482880,     1548288,      86016,      2048;
  7:    7739670528,   10319560704,    6019743744,   2006581248,    418037760,    55738368,    4644864,    221184,     4608;
  8:  103195607040,  154793410560,  103195607040,  40131624960,  10032906240,  1672151040,  185794560,  13271040,   552960,   10240;
  9: 1362182012928, 2270303354880, 1702727516160, 756767784960, 220723937280, 44144787456, 6131220480, 583925760, 36495360, 1351680, 22528;

The row polynomials satisfy (n+1)*P(n,x) = (12*n+24+(2*n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 48 + 8*x.

P[0, x_] := 48 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 24 + (2*n + 4)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 48 + 8*x; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 24 + (2*n + 4)*x)*v[i - 1])/(n + 1)); 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)*2^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 ?

• A147716 → basis A013610 → x ↦ x+1
• A038255: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).
• A013613: Triangle of coefficients in expansion of (1+6x)^n.
• A013613 → D
• A038328 → reverse
• A038255 → D
• A038329 → x ↦ x+1 → x ↦ x+1
• A013612 → x ↦ x/(x+1)
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.
• A038291 → basis A013609 → basis A007318

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.