Real-rooted atlas

A145597: Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.

A145597 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 A145597
\(\left(6\,n + n^{2}\right)\,P_{n}(x) = \left(10 + 10\,x + 9\,n + 9\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2 + 4\,x - 2\,x^{2} - 3\,n + 6\,n x - 3\,n x^{2} - n^{2} + 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
2615627
310454510110
41510518910515429
521210588588210211638
628378151223521512378286188
73663034027560756034026303623256

Show all 15 stored rows

Generated b-files: A003517 (row sums), A145597, A145597 (reversed triangle) ?

Computed in your browser ?

Other recurrences

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

\(\left(6\,n + n^{2}\right)\,P_{n}(x) = \left(12 + 8\,x + 8\,n + 10\,n x + n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,x - 4\,x^{2} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-bb74f03367de3ec8 · revision 2 · JSON
Definition source
oeis_catalog:A145597
derived:reversal (equal by e-d32219abddaaed67)
Fingerprint
f6-cbec7e2198ae346b

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:  3,    3;
  2:  6,   15,     6;
  3: 10,   45,    45,    10;
  4: 15,  105,   189,   105,    15;
  5: 21,  210,   588,   588,   210,    21;
  6: 28,  378,  1512,  2352,  1512,   378,    28;
  7: 36,  630,  3402,  7560,  7560,  3402,   630,    36;
  8: 45,  990,  6930, 20790, 29700, 20790,  6930,   990,   45;
  9: 55, 1485, 13068, 50820, 98010, 98010, 50820, 13068, 1485, 55;
%F Formula
The row polynomials satisfy (n^2+6*n)*P(n,x) = (2*n^2+9*n+10+(2*n^2+9*n+10)*x)*P(n-1,x) - (n^2+3*n+2-(2*n^2+6*n+4)*x+(n^2+3*n+2)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 9*n + 10 + (2*n^2 + 9*n + 10)*x)*P[n - 1, x] - (n^2 + 3*n + 2 - (2*n^2 + 6*n + 4)*x + (n^2 + 3*n + 2)*x^2)*P[n - 2, x])/(n^2 + 6*n)]];
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] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 9*n + 10 + (2*n^2 + 9*n + 10)*x)*v[i - 1] - (n^2 + 3*n + 2 - (2*n^2 + 6*n + 4)*x + (n^2 + 3*n + 2)*x^2)*v[i - 2])/(n^2 + 6*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.

A145597: triangle, offset 2, from the row P0
%e Example
Triangle begins:
   2:  1;
   3:  3,    3;
   4:  6,   15,     6;
   5: 10,   45,    45,    10;
   6: 15,  105,   189,   105,    15;
   7: 21,  210,   588,   588,   210,    21;
   8: 28,  378,  1512,  2352,  1512,   378,    28;
   9: 36,  630,  3402,  7560,  7560,  3402,   630,    36;
  10: 45,  990,  6930, 20790, 29700, 20790,  6930,   990,   45;
  11: 55, 1485, 13068, 50820, 98010, 98010, 50820, 13068, 1485, 55;
%F Formula
The row polynomials satisfy (n^2+2*n-8)*P(n,x) = (2*n^2+n+(2*n^2+n)*x)*P(n-1,x) - (n^2-n-(2*n^2-2*n)*x+(n^2-n)*x^2)*P(n-2,x) for n >= 4, with P(2,x) = 1, P(3,x) = 3 + 3*x.
%t Mathematica
P[2, x_] := 1;
P[3, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n + (2*n^2 + n)*x)*P[n - 1, x] - (n^2 - n - (2*n^2 - 2*n)*x + (n^2 - n)*x^2)*P[n - 2, x])/(n^2 + 2*n - 8)]];
Table[PadRight[CoefficientList[P[n, x], x], n - 1], {n, 2, 11}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 1)); v[1] = 1; v[2] = 3 + 3*x; for(n = 4, N, my(i = n - 1); v[i] = ((2*n^2 + n + (2*n^2 + n)*x)*v[i - 1] - (n^2 - n - (2*n^2 - 2*n)*x + (n^2 - n)*x^2)*v[i - 2])/(n^2 + 2*n - 8)); 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(11)

Generated with polytool c48629374597.

A145597: reversed triangle, offset 2, from the row P0
%e Example
Triangle begins:
   2:  1;
   3:  3,    3;
   4:  6,   15,     6;
   5: 10,   45,    45,    10;
   6: 15,  105,   189,   105,    15;
   7: 21,  210,   588,   588,   210,    21;
   8: 28,  378,  1512,  2352,  1512,   378,    28;
   9: 36,  630,  3402,  7560,  7560,  3402,   630,    36;
  10: 45,  990,  6930, 20790, 29700, 20790,  6930,   990,   45;
  11: 55, 1485, 13068, 50820, 98010, 98010, 50820, 13068, 1485, 55;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n^2+2*n-8)*P(n,x) = (2*n^2+n+(2*n^2+n)*x)*P(n-1,x) - (n^2-n-(2*n^2-2*n)*x+(n^2-n)*x^2)*P(n-2,x) for n >= 4, with P(2,x) = 1, P(3,x) = 3 + 3*x.
%t Mathematica
P[2, x_] := 1;
P[3, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n + (2*n^2 + n)*x)*P[n - 1, x] - (n^2 - n - (2*n^2 - 2*n)*x + (n^2 - n)*x^2)*P[n - 2, x])/(n^2 + 2*n - 8)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n - 1]], {n, 2, 11}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 1)); v[1] = 1; v[2] = 3 + 3*x; for(n = 4, N, my(i = n - 1); v[i] = ((2*n^2 + n + (2*n^2 + n)*x)*v[i - 1] - (n^2 - n - (2*n^2 - 2*n)*x + (n^2 - n)*x^2)*v[i - 2])/(n^2 + 2*n - 8)); 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(11)

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-1670beb9ff565a7b
  • 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-bcb2e3135d28c083
  • 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-bcb2e3135d28c083, e-e8333902099cfe55
  • 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-bcb2e3135d28c083, e-e8333902099cfe55
  • 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-bcb2e3135d28c083
  • 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-e8333902099cfe55

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

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