A038329 → basis A013610 → D+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:            495,             432,               81;
  1:           9450,           13365,             5832,              729;
  2:         172125,          340200,           240570,            69984,             6561;
  3:        3037500,         7745625,          7654500,          3608550,           787320,            59049;
  4:       52396875,       164025000,        209131875,        137781000,         48715425,          8503056,          531441;
  5:      888468750,      3301003125,       5166787500,       4391769375,       2170050750,        613814355,        89282088,         4782969;
  6:    14864765625,     63969750000,     118836112500,     124002900000,      79051848750,      31248730800,      7365772260,       918330048,       43046721;
  7:   246037500000,   1204046015625,    2590774875000,    3208575037500,    2511058725000,    1280639949750,    421857865800,     85232507580,     9298091736,      387420489;
  8:  4036552734375,  22143375000000,   54182070703125,   77723246250000,   72192938343750,   45199057050000,  19209599246250,   5423886846000,   958865710275,    92980917360,   3486784401;
  9: 65738144531250, 399618720703125, 1096097062500000, 1788008333203125, 1923650344687500, 1429420179206250, 745784441325000, 271678617911250, 67120599719250, 10547522813025, 920511081864, 31381059609;

The row polynomials satisfy (3*n+8+3*x)*P(n,x) = (45*n+165+(27*n+144)*x+27*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 495 + 432*x + 81*x^2.

P[0, x_] := 495 + 432*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((45*n + 165 + (27*n + 144)*x + 27*x^2)*P[n - 1, x])/(3*n + 8 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 495 + 432*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((45*n + 165 + (27*n + 144)*x + 27*x^2)*v[i - 1])/(3*n + 8 + 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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^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
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

• A013620 → 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.