Real-rooted atlas

A086645: Triangle read by rows: T(n, k) = binomial(2n, 2k).

A086645 palindromic γ-positive nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0) · coefficients of one sign · interlacing · Hadamard kernel what this means

Recurrence from OEIS A086645
\(P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-1}(x) + \left(-1 + 2\,x - x^{2}\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2112
31618
411515132
512870281128
6145210210451512
71664959244956612048
819110013003300310019118192
91120182080081287080081820120132768
10115330601856443758437581856430601531131072
1111904845387601259701847561259703876048451901524288
12123173157461331977064664664664631977074613731523112097152
131276106261345967354711961256270415619612567354711345961062627618388608
1413251495023023015622755311735965770096577005311735156227523023014950325133554432
15137820475376740310810513123110304217554011660030421755131231103108105376740204753781134217728

Generated b-files: A004171 (row sums), A081294 (row sums), A086645, A086645 (reversed triangle) ?

Computed in your browser ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 - 6\,x + n + 3\,n x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)
Technical details
Node id
n-f7935c36a3c0ec8c · revision 2 · JSON
Definition source
oeis_catalog:A086645
derived:reversal (equal by e-5b61491b47fb76b5)
Fingerprint
f6-3b37e384e5c5a5b6

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

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.

The atlas's own rows: n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,   1;
   3: 1,   6,    1;
   4: 1,  15,   15,     1;
   5: 1,  28,   70,    28,     1;
   6: 1,  45,  210,   210,    45,     1;
   7: 1,  66,  495,   924,   495,    66,     1;
   8: 1,  91, 1001,  3003,  3003,  1001,    91,    1;
   9: 1, 120, 1820,  8008, 12870,  8008,  1820,  120,   1;
  10: 1, 153, 3060, 18564, 43758, 43758, 18564, 3060, 153, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1-2*x+x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 2*x + x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (2 + 2*x)*v[i - 1] - (1 - 2*x + 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, v[i])))};
T(10)

Generated with polytool c48629374597.

A086645: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   6,    1;
  3: 1,  15,   15,     1;
  4: 1,  28,   70,    28,     1;
  5: 1,  45,  210,   210,    45,     1;
  6: 1,  66,  495,   924,   495,    66,     1;
  7: 1,  91, 1001,  3003,  3003,  1001,    91,    1;
  8: 1, 120, 1820,  8008, 12870,  8008,  1820,  120,   1;
  9: 1, 153, 3060, 18564, 43758, 43758, 18564, 3060, 153, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1-2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 2*x + x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 - 2*x + 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)

Generated with polytool c48629374597.

A086645: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   6,    1;
  3: 1,  15,   15,     1;
  4: 1,  28,   70,    28,     1;
  5: 1,  45,  210,   210,    45,     1;
  6: 1,  66,  495,   924,   495,    66,     1;
  7: 1,  91, 1001,  3003,  3003,  1001,    91,    1;
  8: 1, 120, 1820,  8008, 12870,  8008,  1820,  120,   1;
  9: 1, 153, 3060, 18564, 43758, 43758, 18564, 3060, 153, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*x)*P(n-1,x) - (1-2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 2*x + x^2)*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 - 2*x + x^2)*v[i - 2]); 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, 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-6383fa6f6f5b8ab8
  • 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-1375f2f1a11f9809
  • 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-1375f2f1a11f9809, e-a5e3cd5a8b6d6a1a
  • 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-1375f2f1a11f9809, e-a5e3cd5a8b6d6a1a
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-1375f2f1a11f9809
  • 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-a5e3cd5a8b6d6a1a

Proof chains

4 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

Used in 16 proof steps

Same fingerprint (1)

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

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