A391533 → basis A013609 → (θ+1)²

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Submission text ?

Triangle begins:
  0:    9,     48,      36;
  1:   18,    168,     288,      128;
  2:   36,    480,    1332,     1280,      400;
  3:   72,   1248,    4824,     7296,     4800,     1152;
  4:  144,   3072,   15264,    31744,    32400,    16128,     3136;
  5:  288,   7296,   44352,   117760,   164000,   125568,    50176,     8192;
  6:  576,  16896,  121536,   393216,   696000,   723456,   442176,   147456,    20736;
  7: 1152,  38400,  319104,  1218560,  2620800,  3451392,  2853760,  1449984,   414720,    51200;
  8: 2304,  86016,  811008,  3571712,  9049600, 14450688, 15102976, 10354688,  4499712,  1126400,  123904;
  9: 4608, 190464, 2009088, 10027008, 29260800, 54964224, 69543936, 60162048, 35209728, 13363200, 2973696, 294912;

The row polynomials satisfy (9+(27*n+39)*x+(9*n^2+60*n+72)*x^2+(12*n^2+52*n+56)*x^3+(4*n^2+16*n+16)*x^4)*P(n,x) = (18+(54*n+150)*x+(18*n^2+210*n+414)*x^2+(42*n^2+308*n+522)*x^3+(32*n^2+200*n+312)*x^4+(8*n^2+48*n+72)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 48*x + 36*x^2.

P[0, x_] := 9 + 48*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((18 + (54*n + 150)*x + (18*n^2 + 210*n + 414)*x^2 + (42*n^2 + 308*n + 522)*x^3 + (32*n^2 + 200*n + 312)*x^4 + (8*n^2 + 48*n + 72)*x^5)*P[n - 1, x])/(9 + (27*n + 39)*x + (9*n^2 + 60*n + 72)*x^2 + (12*n^2 + 52*n + 56)*x^3 + (4*n^2 + 16*n + 16)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 9 + 48*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((18 + (54*n + 150)*x + (18*n^2 + 210*n + 414)*x^2 + (42*n^2 + 308*n + 522)*x^3 + (32*n^2 + 200*n + 312)*x^4 + (8*n^2 + 48*n + 72)*x^5)*v[i - 1])/(9 + (27*n + 39)*x + (9*n^2 + 60*n + 72)*x^2 + (12*n^2 + 52*n + 56)*x^3 + (4*n^2 + 16*n + 16)*x^4)); 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 The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    • Geometric basis map (elementary) polytool ? step ›

      • from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
                • ↑ Liu–Wang theorem (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
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (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.