A038305 → x ↦ x+1 → (θ+1)²

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:           169,            312,             81;
  1:          2197,           6084,           3159,            432;
  2:         28561,         105456,          82134,          22464,           2025;
  3:        371293,        1713660,        1779570,         730080,         131625,           8748;
  4:       4826809,       26733096,       34701615,       18982080,        5133375,         682344,         35721;
  5:      62748517,      405451956,      631569393,      431842320,      155712375,       31046652,       3250611,        139968;
  6:     815730721,     6023857632,    10947202812,     8982320256,     4048521750,     1076283936,     169031772,      14556672,       531441;
  7:   10604499373,    88098917868,   182974675572,   175155244992,    94735408950,    31481305128,    6592239108,     851565312,     62178597,     1968300;
  8:  137858491849,  1272539924760,  2973338478045,  3252883121280,  2052600527250,   818513933328,  214247771010,   36901163520,   4041608805,   255879000,    7144929;
  9: 1792160394037, 18197320924068, 47243044706715, 58145285792880, 41931696485250, 19507915410984, 6127486250886, 1319216595840, 192650019705, 18295348500, 1021724847, 25509168;

The row polynomials satisfy (169+(117*n+195)*x+(9*n^2+36*n+36)*x^2)*P(n,x) = (2197+(1521*n+4563)*x+(117*n^2+1053*n+1989)*x^2+(27*n^2+162*n+243)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 312*x + 81*x^2.

P[0, x_] := 169 + 312*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2197 + (1521*n + 4563)*x + (117*n^2 + 1053*n + 1989)*x^2 + (27*n^2 + 162*n + 243)*x^3)*P[n - 1, x])/(169 + (117*n + 195)*x + (9*n^2 + 36*n + 36)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 169 + 312*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2197 + (1521*n + 4563)*x + (117*n^2 + 1053*n + 1989)*x^2 + (27*n^2 + 162*n + 243)*x^3)*v[i - 1])/(169 + (117*n + 195)*x + (9*n^2 + 36*n + 36)*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 basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A038327 → basis A013610 → (θ+1)²

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.