A013612 → x ↦ x+1 → (θ+1)²

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Submission text ?

Triangle begins:
  0:        36,         240,          225;
  1:       216,        2160,         4050,         2000;
  2:      1296,       17280,        48600,        48000,         15625;
  3:      7776,      129600,       486000,       720000,        468750,        112500;
  4:     46656,      933120,      4374000,      8640000,       8437500,       4050000,        765625;
  5:    279936,     6531840,     36741600,     90720000,     118125000,      85050000,      32156250,       5000000;
  6:   1679616,    44789760,    293932800,    870912000,    1417500000,    1360800000,     771750000,     240000000,      31640625;
  7:  10077696,   302330880,   2267481600,   7838208000,   15309000000,   18370800000,   13891500000,    6480000000,    1708593750,    195312500;
  8:  60466176,  2015539200,  17006112000,  67184640000,  153090000000,  220449600000,  208372500000,  129600000000,   51257812500,  11718750000,  1181640625;
  9: 362797056, 13302558720, 124711488000, 554273280000, 1443420000000, 2424945600000, 2750517000000, 2138400000000, 1127671875000, 386718750000, 77988281250, 7031250000;

The row polynomials satisfy (36+(90*n+150)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (216+(540*n+1620)*x+(150*n^2+1350*n+2550)*x^2+(125*n^2+750*n+1125)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 240*x + 225*x^2.

P[0, x_] := 36 + 240*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((216 + (540*n + 1620)*x + (150*n^2 + 1350*n + 2550)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*P[n - 1, x])/(36 + (90*n + 150)*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] = 36 + 240*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((216 + (540*n + 1620)*x + (150*n^2 + 1350*n + 2550)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*v[i - 1])/(36 + (90*n + 150)*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 basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.?
    • Geometric basis map (elementary) 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+5x)^n.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A038336 → (θ+1)²
• A038271 → basis A013609 → (θ+1)²
• A038259 → (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.