A123519 → reverse

OEIS matches ?

• A079935: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A001835: 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,     6,     1;
  1:    8,    20,    12,      1;
  2:   16,    56,    60,     20,      1;
  3:   32,   144,   224,    140,     30,      1;
  4:   64,   352,   720,    672,    280,     42,      1;
  5:  128,   832,  2112,   2640,   1680,    504,     56,      1;
  6:  256,  1920,  5824,   9152,   7920,   3696,    840,     72,     1;
  7:  512,  4352, 15360,  29120,  32032,  20592,   7392,   1320,    90,    1;
  8: 1024,  9728, 39168,  87040, 116480,  96096,  48048,  13728,  1980,  110,   1;
  9: 2048, 21504, 97280, 248064, 391680, 396032, 256256, 102960, 24024, 2860, 132, 1;

The row polynomials satisfy (2*n+3)*P(n,x) = (4*n+6+(6*n+7)*x)*P(n-1,x) - (2*x+4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 6*x + x^2.

P[0, x_] := 4 + 6*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 6 + (6*n + 7)*x)*P[n - 1, x] - (2*x + 4*x^2)*D[P[n - 1, x], x])/(2*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 6*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 6 + (6*n + 7)*x)*v[i - 1] - (2*x + 4*x^2)*deriv(v[i - 1], x))/(2*n + 3)); 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-f3eb2c4fabb73e34
• 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-f3eb2c4fabb73e34
• 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-f3eb2c4fabb73e34

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 reversal, shift 0 e-f3eb2c4fabb73e34

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-8d057b494ddd47e8 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

    • theorem_check three_term_recurrence e-2a4594def0262eef root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (2)

Equal up to content, sign, reversal and a small index shift.

• Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0…
• A103327 → basis A013609

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023