A111125 → x ↦ x+1

OEIS matches ?

• A001834: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:
   1:    1;
   2:    4,     1;
   3:   11,     7,     1;
   4:   29,    31,    10,     1;
   5:   76,   115,    60,    13,     1;
   6:  199,   390,   285,    98,    16,     1;
   7:  521,  1254,  1185,   566,   145,    19,    1;
   8: 1364,  3893,  4524,  2785,   985,   201,   22,   1;
   9: 3571, 11789, 16280, 12313,  5595,  1569,  266,  25,  1;
  10: 9349, 35045, 56105, 50434, 28113, 10101, 2345, 340, 28, 1;

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

P[1, x_] := 1;
P[2, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (3 + 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, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-e0cceae8b70995e3
• Coefficients of one sign. Every nonzero row, n >= 1: 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-4d7e15a01b4f3b87, e-e0cceae8b70995e3
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-4d7e15a01b4f3b87, e-e0cceae8b70995e3
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-4d7e15a01b4f3b87, e-e0cceae8b70995e3
• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-4d7e15a01b4f3b87

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-e0cceae8b70995e3

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-0bb8ed02ac535b32 root: theorem check

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

    • theorem_check three_term_recurrence e-19ca7a5e6e7a3cfc root: theorem check

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

• definition_equality e-e62cfe3f46e65c48 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-4d7e15a01b4f3b87

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0)
    • ↵ theorem_check three_term_recurrence e-19ca7a5e6e7a3cfc (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A082985 → x ↦ x/(x+1)

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