A038329 → x ↦ x+1 → (θ+1)²

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Submission text ?

Triangle begins:
  0:           225,            360,              81;
  1:          3375,           8100,            3645,             432;
  2:         50625,         162000,          109350,           25920,            2025;
  3:        759375,        3037500,         2733750,          972000,          151875,           8748;
  4:      11390625,       54675000,        61509375,        29160000,         6834375,         787320,          35721;
  5:     170859375,      956812500,      1291696875,       765450000,       239203125,       41334300,        3750705,        139968;
  6:    2562890625,    16402500000,     25833937500,     18370800000,      7176093750,     1653372000,      225042300,      16796160,       531441;
  7:   38443359375,   276792187500,    498225937500,    413343000000,    193754531250,    55801305000,    10126903500,    1133740800,     71744535,     1968300;
  8:  576650390625,  4613203125000,   9341736328125,   8857350000000,   4843863281250,  1674039150000,   379758881250,   56687040000,   5380840125,   295245000,    7144929;
  9: 8649755859375, 76117851562500, 171265166015625, 182682843750000, 114176777343750, 46036076625000, 12532043081250, 2338340400000, 295946206875, 24357712500, 1178913285, 25509168;

The row polynomials satisfy (25+(15*n+25)*x+(n^2+4*n+4)*x^2)*P(n,x) = (375+(225*n+675)*x+(15*n^2+135*n+255)*x^2+(3*n^2+18*n+27)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 360*x + 81*x^2.

P[0, x_] := 225 + 360*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((375 + (225*n + 675)*x + (15*n^2 + 135*n + 255)*x^2 + (3*n^2 + 18*n + 27)*x^3)*P[n - 1, x])/(25 + (15*n + 25)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 225 + 360*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((375 + (225*n + 675)*x + (15*n^2 + 135*n + 255)*x^2 + (3*n^2 + 18*n + 27)*x^3)*v[i - 1])/(25 + (15*n + 25)*x + (n^2 + 4*n + 4)*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)*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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(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 ?

• A147716 → basis A013610 → (θ+1)²
• A038243 → (θ+1)²
• A038291 → basis A013609 → (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.