A391533 → D

OEIS matches ?

• A113070: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:  4,   2;
  1:  8,  10,    3;
  2: 12,  26,   18,    4;
  3: 16,  50,   57,   28,    5;
  4: 20,  82,  132,  104,   40,    6;
  5: 24, 122,  255,  280,  170,   54,    7;
  6: 28, 170,  438,  620,  520,  258,   70,    8;
  7: 32, 226,  693, 1204, 1295,  882,  371,   88,   9;
  8: 36, 290, 1032, 2128, 2800, 2436, 1400,  512, 108,  10;
  9: 40, 362, 1467, 3504, 5460, 5796, 4242, 2112, 684, 130, 11;

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

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

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

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-0c0d2d375b3b5f35
• 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-0c0d2d375b3b5f35
• 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-0c0d2d375b3b5f35
• 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-0c0d2d375b3b5f35

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-0c0d2d375b3b5f35

  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.