Delannoy paths counted by number of weak peaks.

A133214 (tail) nonnegative coefficients

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

Recurrence from OEIS A133214

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

\(P_{0}(x) = 1 + 8\,x + 4\,x^{2}\)

\(P_{1}(x) = 1 + 18\,x + 36\,x^{2} + 8\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1	8	4								13
1	1	18	36	8							63
2	1	32	144	128	16						321
3	1	50	400	800	400	32					1683
4	1	72	900	3200	3600	1152	64				8989
5	1	98	1764	9800	19600	14112	3136	128			48639
6	1	128	3136	25088	78400	100352	50176	8192	256		265729
7	1	162	5184	56448	254016	508032	451584	165888	20736	512	1462563

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-b31aa1a3fd93d174 · revision 1 · JSON
Definition source: oeis_catalog:A133214
Fingerprint: f6-806d331b1bf3c1f4

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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 ?

A001850: 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1,   8,     4;
  1: 1,  18,    36,      8;
  2: 1,  32,   144,    128,      16;
  3: 1,  50,   400,    800,     400,      32;
  4: 1,  72,   900,   3200,    3600,    1152,       64;
  5: 1,  98,  1764,   9800,   19600,   14112,     3136,      128;
  6: 1, 128,  3136,  25088,   78400,  100352,    50176,     8192,     256;
  7: 1, 162,  5184,  56448,  254016,  508032,   451584,   165888,   20736,     512;
  8: 1, 200,  8100, 115200,  705600, 2032128,  2822400,  1843200,  518400,   51200,   1024;
  9: 1, 242, 12100, 217800, 1742400, 6830208, 13660416, 13939200, 6969600, 1548800, 123904, 2048;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = (2*n+3+(4*n+6)*x)*P(n-1,x) - (n+1-(4*n+4)*x+(4*n+4)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1 + 8*x + 4*x^2, P(1,x) = 1 + 18*x + 36*x^2 + 8*x^3.

%t Mathematica

P[0, x_] := 1 + 8*x + 4*x^2;
P[1, x_] := 1 + 18*x + 36*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 3 + (4*n + 6)*x)*P[n - 1, x] - (n + 1 - (4*n + 4)*x + (4*n + 4)*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 + 8*x + 4*x^2; v[2] = 1 + 18*x + 36*x^2 + 8*x^3; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 3 + (4*n + 6)*x)*v[i - 1] - (n + 1 - (4*n + 4)*x + (4*n + 4)*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.

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-b87132cd573fb9e2
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-4ab8ec5aa6b36e54
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-4ab8ec5aa6b36e54
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-4ab8ec5aa6b36e54
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-4ab8ec5aa6b36e54

Proof chains

2 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

theorem_check three_term_recurrence e-4ab8ec5aa6b36e54 root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
theorem_check three_term_recurrence e-b87132cd573fb9e2 root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Used in 5 proof steps

e-033100947c42f87a basis_map (A008277) → basis_map_A008277 of Delannoy paths counted by number of weak peaks.
e-3165932d1e38479b basis_map (A013609) → basis_map_A013609 of Delannoy paths counted by number of weak peaks.
e-8c219f5be19a6675 basis_map (A013611) → basis_map_A013611 of Delannoy paths counted by number of weak peaks.
e-9a0dabc7f010818b basis_map (A007318) → basis_map_A007318 of Delannoy paths counted by number of weak peaks.
e-f0f0fb352cf0ac03 basis_map (A013610) → basis_map_A013610 of Delannoy paths counted by number of weak peaks.

Related sequences ?

No candidate equalities or related sequences.

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