A172431 → basis A007318

OEIS matches ?

• A060921: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A002450: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:
  0:    1;
  1:    3,     2;
  2:    8,    10,     3;
  3:   21,    38,    22,     4;
  4:   55,   130,   111,    40,     5;
  5:  144,   420,   474,   256,    65,     6;
  6:  377,  1308,  1836,  1324,   511,    98,     7;
  7:  987,  3970,  6666,  6020,  3130,   924,   140,    8;
  8: 2584, 11822, 23109, 25088, 16435,  6588,  1554,  192,   9;
  9: 6765, 34690, 77378, 98160, 77645, 39430, 12720, 2472, 255, 10;

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

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(1 + 2*x + x^2)*P[n - 2, x] + (3 + 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] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + 2*x + x^2)*v[i - 2] + (3 + 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)

Triangle begins:
  0:    1;
  1:    3,     2;
  2:    8,    10,     3;
  3:   21,    38,    22,     4;
  4:   55,   130,   111,    40,     5;
  5:  144,   420,   474,   256,    65,     6;
  6:  377,  1308,  1836,  1324,   511,    98,     7;
  7:  987,  3970,  6666,  6020,  3130,   924,   140,    8;
  8: 2584, 11822, 23109, 25088, 16435,  6588,  1554,  192,   9;
  9: 6765, 34690, 77378, 98160, 77645, 39430, 12720, 2472, 255, 10;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + 2*x + x^2)*v[i - 2] + (3 + 2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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-1a8887aa0d003bc9
• 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-1a8887aa0d003bc9, e-971e090d11ce7802
• 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-1a8887aa0d003bc9, e-971e090d11ce7802
• 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-1a8887aa0d003bc9, e-971e090d11ce7802
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-971e090d11ce7802

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-1a8887aa0d003bc9

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Even row Pascal-square read by antidiagonals.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-83282836b5ac538e root: theorem check

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

• definition_equality e-3fcef085ea71e320 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-971e090d11ce7802

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Even row Pascal-square read by antidiagonals.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ theorem_check three_term_recurrence e-83282836b5ac538e (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A078812 → 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