A013612 → (θ+1)²

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Submission text ?

Triangle begins:
  0: 1,  40,   225;
  1: 1,  60,   675,   2000;
  2: 1,  80,  1350,   8000,   15625;
  3: 1, 100,  2250,  20000,   78125,   112500;
  4: 1, 120,  3375,  40000,  234375,   675000,    765625;
  5: 1, 140,  4725,  70000,  546875,  2362500,   5359375,    5000000;
  6: 1, 160,  6300, 112000, 1093750,  6300000,  21437500,   40000000,   31640625;
  7: 1, 180,  8100, 168000, 1968750, 14175000,  64312500,  180000000,  284765625,   195312500;
  8: 1, 200, 10125, 240000, 3281250, 28350000, 160781250,  600000000, 1423828125,  1953125000,  1181640625;
  9: 1, 220, 12375, 330000, 5156250, 51975000, 353718750, 1650000000, 5220703125, 10742187500, 12998046875, 7031250000;

The row polynomials satisfy (1+(15*n+25)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (1+(15*n+45)*x+(25*n^2+225*n+425)*x^2+(125*n^2+750*n+1125)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 40*x + 225*x^2.

P[0, x_] := 1 + 40*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (15*n + 45)*x + (25*n^2 + 225*n + 425)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*P[n - 1, x])/(1 + (15*n + 25)*x + (25*n^2 + 100*n + 100)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 40*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (15*n + 45)*x + (25*n^2 + 225*n + 425)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*v[i - 1])/(1 + (15*n + 25)*x + (25*n^2 + 100*n + 100)*x^2)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from Triangle of coefficients in expansion of (1+5x)^n.?
    • 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

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.