A013618 → (θ+1)²

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Triangle begins:
  0: 1,  88,  1089;
  1: 1, 132,  3267,   21296;
  2: 1, 176,  6534,   85184,    366025;
  3: 1, 220, 10890,  212960,   1830125,    5797836;
  4: 1, 264, 16335,  425920,   5490375,   34787016,    86806489;
  5: 1, 308, 22869,  745360,  12810875,  121754556,   607645423,   1247178944;
  6: 1, 352, 30492, 1192576,  25621750,  324678816,  2430581692,   9977431552,   17363069361;
  7: 1, 396, 39204, 1788864,  46119150,  730527336,  7291745076,  44898441984,  156267624249,   235794769100;
  8: 1, 440, 49005, 2555520,  76865250, 1461054672, 18229362690, 149661473280,  781338121245,  2357947691000,  3138428376721;
  9: 1, 484, 59895, 3513840, 120788250, 2678600232, 40104597918, 411569051520, 2864906444565, 12968712300500, 34522712143931, 41084880567984;

The row polynomials satisfy (1+(33*n+55)*x+(121*n^2+484*n+484)*x^2)*P(n,x) = (1+(33*n+99)*x+(121*n^2+1089*n+2057)*x^2+(1331*n^2+7986*n+11979)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 88*x + 1089*x^2.

P[0, x_] := 1 + 88*x + 1089*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (33*n + 99)*x + (121*n^2 + 1089*n + 2057)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*P[n - 1, x])/(1 + (33*n + 55)*x + (121*n^2 + 484*n + 484)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 88*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (33*n + 99)*x + (121*n^2 + 1089*n + 2057)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*v[i - 1])/(1 + (33*n + 55)*x + (121*n^2 + 484*n + 484)*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+11x)^n.?
    • 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

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.