A391533 → basis A013609 → D

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:    12,      8;
  1:    42,     64,      24;
  2:   120,    296,     240,      64;
  3:   312,   1072,    1368,     768,     160;
  4:   768,   3392,    5952,    5184,    2240,     384;
  5:  1824,   9856,   22080,   26240,   17440,    6144,     896;
  6:  4224,  27008,   73728,  111360,  100480,   54144,   16128,    2048;
  7:  9600,  70912,  228480,  419328,  479360,  349440,  158592,   40960,    4608;
  8: 21504, 180224,  669696, 1447936, 2007040, 1849344, 1132544,  444416,  101376,  10240;
  9: 47616, 446464, 1880064, 4681728, 7633920, 8515584, 6580224, 3477504, 1202688, 245760, 22528;

The row polynomials satisfy (3*n+1+(2*n+2)*x)*P(n,x) = (6*n+8+(10*n+16)*x+(4*n+8)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 8*x.

P[0, x_] := 12 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 8 + (10*n + 16)*x + (4*n + 8)*x^2)*P[n - 1, x])/(3*n + 1 + (2*n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 12 + 8*x; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 8 + (10*n + 16)*x + (4*n + 8)*x^2)*v[i - 1])/(3*n + 1 + (2*n + 2)*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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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.