narayana → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(32\,x^{2} - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(-2 - 4\,x + 4\,n + 8\,n x\right)\,P_{n-1}(x)\)

\(P_{1}(x) = 1\)

\(P_{2}(x) = 2 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	4							6
3	5	20	16						41
4	14	84	144	64					306
5	42	336	896	896	256				2426
6	132	1320	4800	7680	5120	1024			20076
7	429	5148	23760	52800	57600	27648	4096		171481
8	1430	20020	112112	320320	492800	394240	143360	16384	1500666

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-3af50523900b3357 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-c585b23acdf03520

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: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

A078009: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A127848: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:     2,      4;
   3:     5,     20,      16;
   4:    14,     84,     144,      64;
   5:    42,    336,     896,     896,      256;
   6:   132,   1320,    4800,    7680,     5120,     1024;
   7:   429,   5148,   23760,   52800,    57600,    27648,     4096;
   8:  1430,  20020,  112112,  320320,   492800,   394240,   143360,    16384;
   9:  4862,  77792,  512512, 1793792,  3587584,  4100096,  2523136,   720896,   65536;
  10: 16796, 302328, 2291328, 9504768, 23482368, 35223552, 31309824, 15335424, 3538944, 262144;

%F Formula

The row polynomials satisfy (n+1)*P(n,x) = -(16*n-32)*x^2*P(n-2,x) + (4*n-2+(8*n-4)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n - 32)*x^2*P[n - 2, x] + (4*n - 2 + (8*n - 4)*x)*P[n - 1, 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] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = (-(16*n - 32)*x^2*v[i - 2] + (4*n - 2 + (8*n - 4)*x)*v[i - 1])/(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.

Proved properties

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-e6f59c3f9755ffec
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-e6f59c3f9755ffec
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-e6f59c3f9755ffec
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-e6f59c3f9755ffec

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A013611 e-e6f59c3f9755ffec

Geometric basis map (elementary) polytool ?
- premise (sequence): narayana
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-46a461774e271ec3 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - preserver_image reversal, shift 0 e-29ef477df4172e22
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): narayana
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-46a461774e271ec3 (expanded above)

Related sequences ?

Same fingerprint (1)

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

A001263 → basis A013611

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023