Real-rooted atlas

narayana

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
\(\left(1 + n\right)\,P_{n}(x) = \left(-1 - x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(2 - 4\,x + 2\,x^{2} - n + 2\,n x - n x^{2}\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2112
31315
4166114
51102010142
61155050151132
7121105175105211429
81281964904901962811430
91363361176176411763363614862
10145540252052925292252054045116796
111558254950138601940413860495082555158786
12166121090753267060984609843267090751210661208012
131781716157307078516988422651216988470785157301716781742900
141912366260261431434294297361647361644294291431432602623669112674440
1511053185414052732731002001214714527606152147145100200127327341405318510519694845

Generated b-files: A001263, A001263 (reversed triangle) ?

Computed in your browser ?

Other recurrences

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

\(\left(1 + n\right)\,P_{n}(x) = \left(1 - 3\,x + n + 3\,n x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-2a05d8340c329b21 · revision 2 · JSON
polytool family
narayana
Definition source
named_family:narayana
derived:reversal (equal by e-7615e517905987bb)
Fingerprint
f6-2305441cae17e38c

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,  3,   1;
   4: 1,  6,   6,    1;
   5: 1, 10,  20,   10,    1;
   6: 1, 15,  50,   50,   15,    1;
   7: 1, 21, 105,  175,  105,   21,    1;
   8: 1, 28, 196,  490,  490,  196,   28,   1;
   9: 1, 36, 336, 1176, 1764, 1176,  336,  36,  1;
  10: 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (2*n-1+(2*n-1)*x)*P(n-1,x) - (n-2-(2*n-4)*x+(n-2)*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[Together[((2*n - 1 + (2*n - 1)*x)*P[n - 1, x] - (n - 2 - (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x])/(n + 1)]];
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*n - 1 + (2*n - 1)*x)*v[i - 1] - (n - 2 - (2*n - 4)*x + (n - 2)*x^2)*v[i - 2])/(n + 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)

Generated with polytool c48629374597.

A001263: triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1: 1;
   2: 1,  1;
   3: 1,  3,   1;
   4: 1,  6,   6,    1;
   5: 1, 10,  20,   10,    1;
   6: 1, 15,  50,   50,   15,    1;
   7: 1, 21, 105,  175,  105,   21,    1;
   8: 1, 28, 196,  490,  490,  196,   28,   1;
   9: 1, 36, 336, 1176, 1764, 1176,  336,  36,  1;
  10: 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (2*n-1+(2*n-1)*x)*P(n-1,x) - (n-2-(2*n-4)*x+(n-2)*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[Together[((2*n - 1 + (2*n - 1)*x)*P[n - 1, x] - (n - 2 - (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x])/(n + 1)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {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*n - 1 + (2*n - 1)*x)*v[i - 1] - (n - 2 - (2*n - 4)*x + (n - 2)*x^2)*v[i - 2])/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A001263: reversed triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1: 1;
   2: 1,  1;
   3: 1,  3,   1;
   4: 1,  6,   6,    1;
   5: 1, 10,  20,   10,    1;
   6: 1, 15,  50,   50,   15,    1;
   7: 1, 21, 105,  175,  105,   21,    1;
   8: 1, 28, 196,  490,  490,  196,   28,   1;
   9: 1, 36, 336, 1176, 1764, 1176,  336,  36,  1;
  10: 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n+1)*P(n,x) = (2*n-1+(2*n-1)*x)*P(n-1,x) - (n-2-(2*n-4)*x+(n-2)*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[Together[((2*n - 1 + (2*n - 1)*x)*P[n - 1, x] - (n - 2 - (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x])/(n + 1)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {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*n - 1 + (2*n - 1)*x)*v[i - 1] - (n - 2 - (2*n - 4)*x + (n - 2)*x^2)*v[i - 2])/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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-65e2d844e9e0e220
  • 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-46a461774e271ec3
  • 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-29ef477df4172e22, e-46a461774e271ec3
  • 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-29ef477df4172e22, e-46a461774e271ec3
  • 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-46a461774e271ec3
  • 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-29ef477df4172e22

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 15 proof steps

Same fingerprint (3)

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