A133214 → 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

\(\left(2 + n\right)\,P_{n}(x) = \left(-1 - 16\,x - 64\,x^{2} - n - 16\,n x - 64\,n x^{2}\right)\,P_{n-2}(x) + \left(9 + 24\,x + 6\,n + 16\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 13 + 64\,x + 64\,x^{2}\)

\(P_{1}(x) = 63 + 456\,x + 960\,x^{2} + 512\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	13	64	64															141
1	63	456	960	512														1991
2	321	3072	9984	12288	4096													29761
3	1683	20040	88320	174080	143360	32768												460251
4	8989	128064	713280	1945600	2641920	1572864	262144											7272861
5	48639	806792	5433792	18977280	36270080	36470784	16515072	2097152										116619591
6	265729	5029888	39746560	169394176	419184640	601882624	469762048	167772160	16777216									1889815041
7	1462563	31109256	282166272	1420468224	4323852288	8117157888	9160359936	5737807872	1660944384	134217728								30869546411
8	8097453	191200320	1957700160	11371069440	41131130880	95655297024	142359920640	130358968320	67192750080	16106127360	1073741824							507406003501
9	45046719	1169171784	13338325440	87819072000	368227368960	1023111462912	1898286415872	2310096814080	1757832806400	760343429120	153545080832	8589934592						8382404928711
10	251595969	7119424512	89548791552	659127152640	3144535511040	10171000553472	22656388694016	34570896408576	35203716218880	22677427322880	8362301325312	1443109011456	68719476736					139054141487041
11	1409933619	43199769288	593915961600	4833198729216	25857059205120	95485392617472	248570292731904	457396400947200	587072419135488	509289839984640	281908768407552	89781996355584	13400297963520	549755813888				2314783947556091
12	7923848253	261345892416	3898860044352	34761198710784	206136479797248	856054386130944	2552334557380608	5508290624618496	8560855599808512	9405711016525824	7051864927371264	3395635503955968	944274329829376	123145302310912	4398046511104			38647630102736061
13	44642381823	1576986902280	25371879906240	245956724482560	1601445853163520	7389404805758976	24841405798809600	61581646312243200	112621538141798400	150396926846566400	143439511337041920	94048316219719680	39803351717642240	9755416917442560	1121501860331520	35184372088832		646908600416279751
14	252055236609	9494323789824	163866548367360	1716135321600000	12172130717859840	61793056050905088	231324627467501568	648172805082316800	1365337973352038400	2152087852941312000	2503546401162526720	2096664220564193280	1214174197878620160	455637618550374400	99219929290506240	10133099161583616	281474976710656	10852435135445442561

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Technical details

Node id: n-4acec5e35bbb846a · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-cbb4e317a78480a3

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 ?

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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:       13,         64,          64;
  1:       63,        456,         960,         512;
  2:      321,       3072,        9984,       12288,         4096;
  3:     1683,      20040,       88320,      174080,       143360,         32768;
  4:     8989,     128064,      713280,     1945600,      2641920,       1572864,        262144;
  5:    48639,     806792,     5433792,    18977280,     36270080,      36470784,      16515072,       2097152;
  6:   265729,    5029888,    39746560,   169394176,    419184640,     601882624,     469762048,     167772160,      16777216;
  7:  1462563,   31109256,   282166272,  1420468224,   4323852288,    8117157888,    9160359936,    5737807872,    1660944384,    134217728;
  8:  8097453,  191200320,  1957700160, 11371069440,  41131130880,   95655297024,  142359920640,  130358968320,   67192750080,  16106127360,   1073741824;
  9: 45046719, 1169171784, 13338325440, 87819072000, 368227368960, 1023111462912, 1898286415872, 2310096814080, 1757832806400, 760343429120, 153545080832, 8589934592;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = -(n+1+(16*n+16)*x+(64*n+64)*x^2)*P(n-2,x) + (6*n+9+(16*n+24)*x)*P(n-1,x) for n >= 2, with P(0,x) = 13 + 64*x + 64*x^2, P(1,x) = 63 + 456*x + 960*x^2 + 512*x^3.

%t Mathematica

P[0, x_] := 13 + 64*x + 64*x^2;
P[1, x_] := 63 + 456*x + 960*x^2 + 512*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (16*n + 16)*x + (64*n + 64)*x^2)*P[n - 2, x] + (6*n + 9 + (16*n + 24)*x)*P[n - 1, 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] = 13 + 64*x + 64*x^2; v[2] = 63 + 456*x + 960*x^2 + 512*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 1 + (16*n + 16)*x + (64*n + 64)*x^2)*v[i - 2] + (6*n + 9 + (16*n + 24)*x)*v[i - 1])/(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, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-8c219f5be19a6675
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-8c219f5be19a6675
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-8c219f5be19a6675
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-8c219f5be19a6675

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-8c219f5be19a6675

Geometric basis map (elementary) polytool ?
- premise (sequence): Delannoy paths counted by number of weak peaks.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-4ab8ec5aa6b36e54 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

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