narayana → basis A008277

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	1							3
3	5	6	1						12
4	14	31	12	1					58
5	42	155	105	20	1				323
6	132	771	815	265	30	1			2014
7	429	3850	5971	2940	560	42	1		13793
8	1430	19363	42420	29631	8400	1050	56	1	102351

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-a91ed9a3e111e514 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-c00a555c8a33253e

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 ?

No OEIS matches.

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,      1;
   3:     5,      6,       1;
   4:    14,     31,      12,       1;
   5:    42,    155,     105,      20,       1;
   6:   132,    771,     815,     265,      30,      1;
   7:   429,   3850,    5971,    2940,     560,     42,     1;
   8:  1430,  19363,   42420,   29631,    8400,   1050,    56,    1;
   9:  4862,  98187,  296269,  281862,  111195,  20454,  1806,   72,  1;
  10: 16796, 502063, 2049855, 2583715, 1359792, 343749, 44310, 2910, 90, 1;

%F Formula

The row polynomials satisfy (n+1)*P(n,x) = -((n-2)*x+(n-2)*x^2)*P(n-2,x) - ((n-2)*x+(2*n-4)*x^2)*d/dx P(n-2,x) - (n-2)*x^2*d^2/dx^2 P(n-2,x) + (4*n-2+(2*n-1)*x)*P(n-1,x) + (2*n-1)*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-((n - 2)*x + (n - 2)*x^2)*P[n - 2, x] - ((n - 2)*x + (2*n - 4)*x^2)*D[P[n - 2, x], x] - (n - 2)*x^2*D[P[n - 2, x], {x, 2}] + (4*n - 2 + (2*n - 1)*x)*P[n - 1, x] + (2*n - 1)*x*D[P[n - 1, x], 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 + x; for(n = 3, N, my(i = n); v[i] = (-((n - 2)*x + (n - 2)*x^2)*v[i - 2] - ((n - 2)*x + (2*n - 4)*x^2)*deriv(v[i - 2], x) - (n - 2)*x^2*deriv(deriv(v[i - 2], x), x) + (4*n - 2 + (2*n - 1)*x)*v[i - 1] + (2*n - 1)*x*deriv(v[i - 1], x))/(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-9b44e0c74e7ea693
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-9b44e0c74e7ea693
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-9b44e0c74e7ea693

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 A008277 e-9b44e0c74e7ea693

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- 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 A008277

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023