A391533 → D+1

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:  8,   6,    1;
  1: 12,  18,    8,    1;
  2: 16,  38,   31,   10,    1;
  3: 20,  66,   82,   47,   12,    1;
  4: 24, 102,  173,  148,   66,   14,    1;
  5: 28, 146,  316,  365,  240,   88,   16,    1;
  6: 32, 198,  523,  766,  675,  362,  113,   18,   1;
  7: 36, 258,  806, 1435, 1596, 1141,  518,  141,  20,   1;
  8: 40, 326, 1177, 2472, 3332, 2996, 1806,  712, 172,  22,  1;
  9: 44, 402, 1648, 3993, 6336, 6888, 5208, 2718, 948, 206, 24, 1;

The row polynomials satisfy (2*n+2+(n+4)*x+x^2)*P(n,x) = (2*n+4+(3*n+9)*x+(n+6)*x^2+x^3)*P(n-1,x) for n >= 1, with P(0,x) = 8 + 6*x + x^2.

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

P(N) = {my(v = vector(N + 1)); v[1] = 8 + 6*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 4 + (3*n + 9)*x + (n + 6)*x^2 + x^3)*v[i - 1])/(2*n + 2 + (n + 4)*x + x^2)); 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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-bce9a2f56a3e94bb
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-bce9a2f56a3e94bb
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-bce9a2f56a3e94bb
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-bce9a2f56a3e94bb

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• preserver_image derivative_polynomial, shift 0 e-bce9a2f56a3e94bb

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): 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.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-3cff49c1888b5e80 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers periodic_product e-5e5f630c1e2baa8a root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

1. [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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.