A391533 → reverse → D

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:  4,   8;
  1:  5,  16,  12;
  2:  6,  26,  36,   16;
  3:  7,  38,  75,   64,   20;
  4:  8,  52, 132,  164,  100,   24;
  5:  9,  68, 210,  340,  305,  144,   28;
  6: 10,  86, 312,  620,  730,  510,  196,   32;
  7: 11, 106, 441, 1036, 1505, 1386,  791,  256,   36;
  8: 12, 128, 600, 1624, 2800, 3192, 2408, 1160,  324,  40;
  9: 13, 152, 792, 2424, 4830, 6552, 6132, 3912, 1629, 400, 44;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 8*x; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (3*n + 8)*x + (2*n + 4)*x^2)*v[i - 1])/(n + 3 + (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 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 [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.?
            no recorded property covers this requirement
        • 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

Related ?

Same fingerprint ?

• A159854 → basis A013609 → D

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.