Real-rooted atlas

A145596: basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x -> x + 1

A145596 palindromic γ-positive nonnegative coefficients

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

Recurrence
\(\left(4\,n + n^{2}\right)\,P_{n}(x) = \left(4 + 2\,x + 5\,n + 5\,n x + n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
238314
342020448
454075405165
5670210210706572
6711249078449011272002
78168100823522352100816887072

Show all 15 stored rows

Generated b-files: A002057 (row sums), A145596, A145596 (reversed triangle) ?

Computed in your browser ?

Other recurrences

OEIS A145596 (OEIS) · proved equal for all n (proof step)

\(\left(4 + n\right)\,P_{n}(x) = \left(7 + 7\,x + 3\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-2 - 2\,x^{2} - 3\,n - 2\,n x - 3\,n x^{2}\right)\,P_{n-2}(x) + \left(-1 + x + x^{2} - x^{3} + n - n x - n x^{2} + n x^{3}\right)\,P_{n-3}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)
\(P_{2}(x) = 3 + 8\,x + 3\,x^{2}\)
Technical details
Node id
n-0e6d847331141ca6 · revision 2 · JSON
Definition source
derived:root_bound_shift:derived:basis_map_coefficients_of:oeis_catalog:A145596
oeis_catalog:A145596 (equal by e-34496f1b474c1a7e)
Fingerprint
f6-f1789e1ed3fa2160

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  2,   2;
  2:  3,   8,    3;
  3:  4,  20,   20,     4;
  4:  5,  40,   75,    40,     5;
  5:  6,  70,  210,   210,    70,     6;
  6:  7, 112,  490,   784,   490,   112,     7;
  7:  8, 168, 1008,  2352,  2352,  1008,   168,    8;
  8:  9, 240, 1890,  6048,  8820,  6048,  1890,  240,   9;
  9: 10, 330, 3300, 13860, 27720, 27720, 13860, 3300, 330, 10;
%F Formula
The row polynomials satisfy (n^2+4*n)*P(n,x) = (n^2+5*n+4+(3*n^2+5*n+2)*x)*P(n-1,x) + ((2*n+2)*x-(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 5*n + 4 + (3*n^2 + 5*n + 2)*x)*P[n - 1, x] + ((2*n + 2)*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 5*n + 4 + (3*n^2 + 5*n + 2)*x)*v[i - 1] + ((2*n + 2)*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n)); 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)

Generated with polytool c48629374597.

A145596: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:  1;
   2:  2,   2;
   3:  3,   8,    3;
   4:  4,  20,   20,     4;
   5:  5,  40,   75,    40,     5;
   6:  6,  70,  210,   210,    70,     6;
   7:  7, 112,  490,   784,   490,   112,     7;
   8:  8, 168, 1008,  2352,  2352,  1008,   168,    8;
   9:  9, 240, 1890,  6048,  8820,  6048,  1890,  240,   9;
  10: 10, 330, 3300, 13860, 27720, 27720, 13860, 3300, 330, 10;
%F Formula
The row polynomials satisfy (n^2+2*n-3)*P(n,x) = (n^2+3*n+(3*n^2-n)*x)*P(n-1,x) + (2*n*x-2*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 3*n + (3*n^2 - n)*x)*P[n - 1, x] + (2*n*x - 2*n*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n - 3)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^2 + 3*n + (3*n^2 - n)*x)*v[i - 1] + (2*n*x - 2*n*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n - 3)); 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.

A145596: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:  1;
   2:  2,   2;
   3:  3,   8,    3;
   4:  4,  20,   20,     4;
   5:  5,  40,   75,    40,     5;
   6:  6,  70,  210,   210,    70,     6;
   7:  7, 112,  490,   784,   490,   112,     7;
   8:  8, 168, 1008,  2352,  2352,  1008,   168,    8;
   9:  9, 240, 1890,  6048,  8820,  6048,  1890,  240,   9;
  10: 10, 330, 3300, 13860, 27720, 27720, 13860, 3300, 330, 10;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n^2+2*n-3)*P(n,x) = (n^2+3*n+(3*n^2-n)*x)*P(n-1,x) + (2*n*x-2*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 3*n + (3*n^2 - n)*x)*P[n - 1, x] + (2*n*x - 2*n*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n - 3)]];
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; for(n = 2, N, my(i = n); v[i] = ((n^2 + 3*n + (3*n^2 - n)*x)*v[i - 1] + (2*n*x - 2*n*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n - 3)); 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 >= 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-c483eeee9c2638c9
  • 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-1d2c7b4a3cb3d72f, e-c483eeee9c2638c9, e-fb63394f9047a486
  • 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-1d2c7b4a3cb3d72f, e-c483eeee9c2638c9, e-fb63394f9047a486
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-c483eeee9c2638c9
  • 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-1d2c7b4a3cb3d72f, e-fb63394f9047a486
  • 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-fb63394f9047a486
  • 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-239cba93e63d8d6d

Proof chains

5 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 10 proof steps

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023