A123519 → x ↦ x+1

OEIS matches ?

• A001653: 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:      11,      14,        4;
  1:      41,      76,       44,        8;
  2:     153,     372,      324,      120,       16;
  3:     571,    1718,     1996,     1120,      304,       32;
  4:    2131,    7642,    11096,     8352,     3440,      736,       64;
  5:    7953,   33112,    57672,    54480,    30160,     9792,     1728,     128;
  6:   29681,  140712,   285816,   324912,   226160,    98752,    26432,    3968,     256;
  7:  110771,  589098,  1367016,  1816800,  1524304,   837536,   301504,   68608,    8960,    512;
  8:  413403, 2437222,  6360444,  9676320,  9504656,  6300000,  2854656,  873472,  172800,  19968,  1024;
  9: 1542841, 9986596, 28949204, 49609368, 55846960, 43371776, 23717120, 9134592, 2429184, 424960, 44032, 2048;

The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 11 + 14*x + 4*x^2, P(1,x) = 41 + 76*x + 44*x^2 + 8*x^3.

P[0, x_] := 11 + 14*x + 4*x^2;
P[1, x_] := 41 + 76*x + 44*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 11 + 14*x + 4*x^2; v[2] = 41 + 76*x + 44*x^2 + 8*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (4 + 2*x)*v[i - 1]); 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-fe32eb1bdf118767
• 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-762ccb2ce32a5929, e-fe32eb1bdf118767
• 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-762ccb2ce32a5929, e-fe32eb1bdf118767
• 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-762ccb2ce32a5929, e-fe32eb1bdf118767
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-3, -1)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-762ccb2ce32a5929

Proof chains

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

• basis_map A007318 e-fe32eb1bdf118767

  Geometric basis map (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 ?

• definition_equality e-cd46c8ba2133c8f0 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-762ccb2ce32a5929

  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 (-2, 0)
    • ↵ theorem_check three_term_recurrence e-2a4594def0262eef (expanded above)

Related sequences ?

No candidate equalities or related sequences.

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