Real-rooted atlas

A001263: Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.

A001263 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 A001263
\(\left(2 + n\right)\,P_{n}(x) = \left(1 + x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(1 - 2\,x + x^{2} - n + 2\,n x - n x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21315
3166114
41102010142
51155050151132
6121105175105211429
71281964904901962811430

Show all 15 stored rows

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

Computed in your browser ?

Other recurrences

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

\(\left(2 + n\right)\,P_{n}(x) = \left(2 + n + 3\,n x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-c2eaf5212a4dec13 · revision 2 · JSON
Definition source
oeis_catalog:A001263
derived:reversal (equal by e-5f2e71a1e47d5fe3)
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 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: 1,  1;
  2: 1,  3,   1;
  3: 1,  6,   6,    1;
  4: 1, 10,  20,   10,    1;
  5: 1, 15,  50,   50,   15,    1;
  6: 1, 21, 105,  175,  105,   21,    1;
  7: 1, 28, 196,  490,  490,  196,   28,   1;
  8: 1, 36, 336, 1176, 1764, 1176,  336,  36,  1;
  9: 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (2*n+1+(2*n+1)*x)*P(n-1,x) - (n-1-(2*n-2)*x+(n-1)*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[Together[((2*n + 1 + (2*n + 1)*x)*P[n - 1, x] - (n - 1 - (2*n - 2)*x + (n - 1)*x^2)*P[n - 2, x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {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*n + 1 + (2*n + 1)*x)*v[i - 1] - (n - 1 - (2*n - 2)*x + (n - 1)*x^2)*v[i - 2])/(n + 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)

Generated with polytool c48629374597.

A001263: triangle, offset 1, from the row P0
%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 P0
%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 >= 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-fa7c13a12b3db82b
  • 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-d285019b29e04549
  • 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-14da04f2a2dce119, e-d285019b29e04549
  • 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-14da04f2a2dce119, e-d285019b29e04549
  • 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-d285019b29e04549
  • 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-14da04f2a2dce119

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 13 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