A013612 → basis A013609 → θ+1

Matches ?

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Submission text ?

Triangle begins:
  0:        36,         240,          300;
  1:       216,        2160,         5400,          4000;
  2:      1296,       17280,        64800,         96000,         50000;
  3:      7776,      129600,       648000,       1440000,       1500000,         600000;
  4:     46656,      933120,      5832000,      17280000,      27000000,       21600000,        7000000;
  5:    279936,     6531840,     48988800,     181440000,     378000000,      453600000,      294000000,       80000000;
  6:   1679616,    44789760,    391910400,    1741824000,    4536000000,     7257600000,     7056000000,     3840000000,      900000000;
  7:  10077696,   302330880,   3023308800,   15676416000,   48988800000,    97977600000,   127008000000,   103680000000,    48600000000,    10000000000;
  8:  60466176,  2015539200,  22674816000,  134369280000,  489888000000,  1175731200000,  1905120000000,  2073600000000,  1458000000000,   600000000000,  110000000000;
  9: 362797056, 13302558720, 166281984000, 1108546560000, 4618944000000, 12933043200000, 25147584000000, 34214400000000, 32076000000000, 19800000000000, 7260000000000, 1200000000000;

The row polynomials satisfy (3+(5*n+10)*x)*P(n,x) = (18+(30*n+120)*x+(50*n+150)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 240*x + 300*x^2.

P[0, x_] := 36 + 240*x + 300*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((18 + (30*n + 120)*x + (50*n + 150)*x^2)*P[n - 1, x])/(3 + (5*n + 10)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 36 + 240*x + 300*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((18 + (30*n + 120)*x + (50*n + 150)*x^2)*v[i - 1])/(3 + (5*n + 10)*x)); 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_A013609 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

Related ?

Same fingerprint ?

• A013622 → θ+1
• A038271 → basis A013611 → θ+1

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.