A103327 → basis A007318

OEIS matches ?

• A373547: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A001653: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

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:      4,       1;
  2:     16,      12,       1;
  3:     64,      80,      24,       1;
  4:    256,     448,     240,      40,       1;
  5:   1024,    2304,    1792,     560,      60,      1;
  6:   4096,   11264,   11520,    5376,    1120,     84,     1;
  7:  16384,   53248,   67584,   42240,   13440,   2016,   112,    1;
  8:  65536,  245760,  372736,  292864,  126720,  29568,  3360,  144,   1;
  9: 262144, 1114112, 1966080, 1863680, 1025024, 329472, 59136, 5280, 180, 1;

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

P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[-x^2*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] = 1; v[2] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = -x^2*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)

Triangle begins:
  0: 1;
  1: 1,   4;
  2: 1,  12,   16;
  3: 1,  24,   80,    64;
  4: 1,  40,  240,   448,    256;
  5: 1,  60,  560,  1792,   2304,    1024;
  6: 1,  84, 1120,  5376,  11520,   11264,    4096;
  7: 1, 112, 2016, 13440,  42240,   67584,   53248,   16384;
  8: 1, 144, 3360, 29568, 126720,  292864,  372736,  245760,   65536;
  9: 1, 180, 5280, 59136, 329472, 1025024, 1863680, 1966080, 1114112, 262144;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = -x^2*P(n-2,x) + (4+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] + (4 + 2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(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-6f2b6c095b5c4221
• 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-6f2b6c095b5c4221, e-dbf43cf157e06eb8
• 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-6f2b6c095b5c4221, e-dbf43cf157e06eb8
• 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-6f2b6c095b5c4221, e-dbf43cf157e06eb8
• 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-dbf43cf157e06eb8

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-6f2b6c095b5c4221

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-86d927bb13c60f05 root: theorem check

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

• definition_equality e-19e7de6d44e2d2c6 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-dbf43cf157e06eb8

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ theorem_check three_term_recurrence e-86d927bb13c60f05 (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A091042 → 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