A038328 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(22 + 16\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 484 + 704\,x + 256\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	484	704	256															1444
1	10648	23232	16896	4096														54872
2	234256	681472	743424	360448	65536													2085136
3	5153632	18740480	27258880	19824640	7208960	1048576												79235168
4	113379904	494748672	899543040	872284160	475791360	138412032	16777216											3010936384
5	2494357888	12698549248	27705925632	33582940160	24423956480	10657726464	2583691264	268435456										114415582592
6	54875873536	319277809664	812707151872	1182119493632	1074654085120	625253285888	227364831232	47244640256	4294967296									4347792138496
7	1207269217792	7902125789184	22988002295808	39009943289856	42556301770752	30950037651456	15006078861312	4677219385344	850403524608	68719476736								165216101262848
8	26559922791424	193163074846720	632170063134720	1226026789109760	1560397731594240	1361801656664064	825334337372160	342996088258560	93544387706880	15118284881920	1099511627776							6278211847988224
9	584318301411328	4674546411290624	16998350586511360	37087310370570240	53945178720829440	54926000152117248	39946181928812544	20751263339642880	7545913941688320	1829312470712320	266081813921792	17592186044416						238572050223552512
10	12855002631049216	112189113870974976	448756455483899904	1087894437536727040	1780190897787371520	2071494862879850496	1757632004867751936	1095666704333144064	498030320151429120	160979497422684160	35122799437676544	4644337115725824	281474976710656					9065737908494995456
11	282810057883082752	2673840547258236928	11667667842581397504	31113780913550393344	56570510751909806080	74055941347954655232	71811821913168150528	52226779573213200384	28487334312661745664	11510034065721917440	3348373546391830528	664140207548792832	80501843339247616	4503599627370496				344498040522809827328
12	6221821273427820544	63349452965810536448	299470141292922535936	871185865579411013632	1742371731158822027264	2534358881685559312384	2764755143656973795328	2297978301221380816896	1462349828049969610752	709018098448470114304	257824763072170950656	68185061308342730752	12397283874244132864	1387108685230112768	72057594037927936			13090925539866773438464
13	136880068015412051968	1493237105622676930560	7601934355897264373760	23957611303433802874880	52271151934764660817920	83633843095623457308672	101374355267422372495360	94791604925381958696960	68939349036641424506880	38995995414665856286720	17016434362763282743296	5625267557938275287040	1363701226166854615040	228872933062968606720	23779006032516218880	1152921504606846976		497455170514937390661632
14	3011361496339065143296	35041297411945485303808	191134349519702647111680	648698398369899893227520	1533287123419763383992320	2676282979059950633877504	3568377305413267511836672	3707404992637161051258880	3033331357612222678302720	1960941483708911630417920	998297482615445920940032	396018836078854580207616	120005707902683206123520	26854424146054983188480	4185105061722854522880	405828369621610135552	18446744073709551616	18903296479567620845142016

Technical details

Node id: n-e6ce82432a2d62f1 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-e348a50f4e54dede

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:             484,              704,               256;
  1:           10648,            23232,             16896,              4096;
  2:          234256,           681472,            743424,            360448,             65536;
  3:         5153632,         18740480,          27258880,          19824640,           7208960,           1048576;
  4:       113379904,        494748672,         899543040,         872284160,         475791360,         138412032,          16777216;
  5:      2494357888,      12698549248,       27705925632,       33582940160,       24423956480,       10657726464,        2583691264,         268435456;
  6:     54875873536,     319277809664,      812707151872,     1182119493632,     1074654085120,      625253285888,      227364831232,       47244640256,       4294967296;
  7:   1207269217792,    7902125789184,    22988002295808,    39009943289856,    42556301770752,    30950037651456,    15006078861312,     4677219385344,     850403524608,      68719476736;
  8:  26559922791424,  193163074846720,   632170063134720,  1226026789109760,  1560397731594240,  1361801656664064,   825334337372160,   342996088258560,   93544387706880,   15118284881920,   1099511627776;
  9: 584318301411328, 4674546411290624, 16998350586511360, 37087310370570240, 53945178720829440, 54926000152117248, 39946181928812544, 20751263339642880, 7545913941688320, 1829312470712320, 266081813921792, 17592186044416;

%F Formula

The row polynomials satisfy P(n,x) = (22+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 704*x + 256*x^2.

%t Mathematica

P[0, x_] := 484 + 704*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 16*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 484 + 704*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 16*x)*v[i - 1]); 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.