A038319 → basis A013610 → θ+1

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Triangle begins:
  0:            256,             960,              675;
  1:           4096,           23040,            32400,            13500;
  2:          65536,          491520,          1036800,           864000,            253125;
  3:        1048576,         9830400,         27648000,         34560000,          20250000,           4556250;
  4:       16777216,       188743680,        663552000,       1105920000,         972000000,         437400000,          79734375;
  5:      268435456,      3523215360,      14863564800,      30965760000,       36288000000,       24494400000,        8930250000,        1366875000;
  6:     4294967296,     64424509440,     317089382400,     792723456000,     1161216000000,     1045094400000,      571536000000,      174960000000,       23066015625;
  7:    68719476736,   1159641169920,    6522981580800,   19025362944000,    33443020800000,    37623398400000,    27433728000000,    12597120000000,     3321506250000,     384433593750;
  8:  1099511627776,  20615843020800,  130459631616000,  434865438720000,   891813888000000,  1203948748800000,  1097349120000000,   671846400000000,   265720500000000,   61509375000000,    6343154296875;
  9: 17592186044416, 362838837166080, 2551210573824000, 9567039651840000, 22422749184000000, 35315829964800000, 38626689024000000, 29561241600000000, 15588936000000000, 5412825000000000, 1116395156250000, 103797070312500;

The row polynomials satisfy (16+(15*n+30)*x)*P(n,x) = (256+(240*n+960)*x+(225*n+675)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 960*x + 675*x^2.

P[0, x_] := 256 + 960*x + 675*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((256 + (240*n + 960)*x + (225*n + 675)*x^2)*P[n - 1, x])/(16 + (15*n + 30)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 256 + 960*x + 675*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((256 + (240*n + 960)*x + (225*n + 675)*x^2)*v[i - 1])/(16 + (15*n + 30)*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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^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
        • Geometric basis map (elementary) 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)
        • 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 basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.?
            • ↑ Geometric basis map (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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.