A085841: Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ).

OEIS matches ?

• A085841: triangle, 36 terms agree, 8 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A066443: 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:
  0:  1;
  1:  3,    4;
  2:  5,   40,     16;
  3:  7,  140,    336,      64;
  4:  9,  336,   2016,    2304,      256;
  5: 11,  660,   7392,   21120,    14080,     1024;
  6: 13, 1144,  20592,  109824,   183040,    79872,      4096;
  7: 15, 1820,  48048,  411840,  1281280,  1397760,    430080,    16384;
  8: 17, 2720,  99008, 1244672,  6223360, 12673024,   9748480,  2228224,    65536;
  9: 19, 3876, 186048, 3224832, 23648768, 77395968, 111132672, 63504384, 11206656, 262144;

The row polynomials satisfy P(n,x) = (2+8*x)*P(n-1,x) - (1-8*x+16*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[(2 + 8*x)*P[n - 1, x] - (1 - 8*x + 16*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 8*x)*v[i - 1] - (1 - 8*x + 16*x^2)*v[i - 2]); 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,    4;
  2:  5,   40,     16;
  3:  7,  140,    336,      64;
  4:  9,  336,   2016,    2304,      256;
  5: 11,  660,   7392,   21120,    14080,     1024;
  6: 13, 1144,  20592,  109824,   183040,    79872,      4096;
  7: 15, 1820,  48048,  411840,  1281280,  1397760,    430080,    16384;
  8: 17, 2720,  99008, 1244672,  6223360, 12673024,   9748480,  2228224,    65536;
  9: 19, 3876, 186048, 3224832, 23648768, 77395968, 111132672, 63504384, 11206656, 262144;

The row polynomials satisfy P(n,x) = (2+8*x)*P(n-1,x) - (1-8*x+16*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[(2 + 8*x)*P[n - 1, x] - (1 - 8*x + 16*x^2)*P[n - 2, 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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 8*x)*v[i - 1] - (1 - 8*x + 16*x^2)*v[i - 2]); 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, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-031667aece1ffd0c
• 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-c0dcd56725ae4481
• 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-c0dcd56725ae4481
• 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-c0dcd56725ae4481
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-c0dcd56725ae4481

Proof chains

2 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

• theorem_check three_term_recurrence e-031667aece1ffd0c root: theorem check

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

• theorem_check three_term_recurrence e-c0dcd56725ae4481 root: theorem check

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

Used in 5 proof steps

• e-51fc7cec3579afbb basis_map (A013610) → basis_map_A013610 of Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2…
• e-8c1265744c32866c basis_map (A007318) → basis_map_A007318 of Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2…
• e-b4660546a5e77fa6 basis_map (A013609) → basis_map_A013609 of Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2…
• e-cc801bbe6c76dcc1 basis_map (A013611) → basis_map_A013611 of Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2…
• e-db4d4459bbf9061f basis_map (A008277) → basis_map_A008277 of Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2…

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