A038334 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(44 + 72\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1936 + 6336\,x + 5184\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1936	6336	5184															13456
1	85184	418176	684288	373248														1560896
2	3748096	24532992	60217344	65691648	26873856													181063936
3	164916224	1349314560	4415938560	7226081280	5912248320	1934917632												21003416576
4	7256313856	71243808768	291451944960	635895152640	780416778240	510818254848	139314069504											2436396322816
5	319277809664	3657182183424	17953439809536	48963926753280	80122789232640	78666011246592	42908733407232	10030613004288										282621973446656
6	14048223625216	183904018366464	1053268468826112	3447060443430912	7050805452472320	9230145319600128	7551937079672832	3530775777509376	722204136308736									32784148919812096
7	618121839509504	9103248909139968	59584901950734336	227505989266440192	558423791835807744	913784386640412672	996855694516813824	699093603946856448	285992837978259456	51998697814228992								3802961274698203136
8	27197360938418176	445047724446842880	3277169607290388480	14300376468176240640	40951078067959234560	80413026024356315136	109654126396849520640	102533728578872279040	62918424355217080320	22879427038260756480	3743906242624487424							441143507864991563776
9	1196683881290399744	21540309863227195392	176238898880949780480	865172776324662558720	2831474540698895646720	6486650765964742754304	10614519435215033597952	12406581158043545763840	10150839129308355624960	5536821343259103068160	1812050621430251913216	269561249468963094528						51172646912339021398016
10	52654090776777588736	1033934873434905378816	9305413860914148409344	50756802877713536778240	186877319686127112683520	489278800632769167753216	934077710298922956619776	1310134970289398432661504	1339910765068702942494720	974480556413602139996160	478381364057586505089024	142328339719612513910784	19408409961765342806016					5936027041831326482169856
11	2316779994178213904384	49284228967063823056896	483881520767535717285888	2903289124605214303715328	11877091873384967606108160	34983434245242995494354944	76327492898711990169501696	124899533834255983913730048	153285791523859616621395968	139350719567145106019450880	91211380080313160303640576	40705905159809178978484224	11101610498129776085041152	1397405517247104682033152				688579136852433871931703296
12	101938319743841411792896	2335314234131639615619072	24839251399400166820675584	162584190977892001008058368	731628859400514004536262656	2394421721674409469391405056	5877216953200823243051630592	10991158977414526584408244224	15737341263116253973129986048	17168008650672277061596348416	14046552532368226686760648704	8358279192814151416915427328	3419296033423971034192674816	860801798624216484132421632	100613197241791537106386944			79875179874882329144077582336
13	4485286068729022118887424	110093385323348724736327680	1261069686431085392434298880	8942130503784060055443210240	43897731564030840272175759360	158031833630511024979832733696	430995909901393704490452910080	906770615636698443213680148480	1483806461950961088895112970240	1888480951573950476775598325760	1854144934272605922652405628928	1379116066814334983791045509120	752245127353273627522388459520	284064593545991439763699138560	66404710179582414490215383040	7244150201408990671659859968		9265520865486350180712999550976
14	197352587024076973231046656	5167049551175833480958312448	63413789946248865448124743680	484250759589536790694770769920	2575333585089809295967644549120	10114037352352705598709294956544	30342112057058116796127884869632	70929612600915078224714536058880	130574968651684575822769941381120	189927227129723019378574460190720	217553005621319094924548927127552	194179542207458365717779207684096	132395142414176158443940368875520	66660491285459324531214731182080	23374457983213009900555814830080	5099881741791929432848541417472	521578814501447328359509917696	1074800420396416620962707947913216

Technical details

Node id: n-1f571e6a0df94da8 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-5e3baf07c55bade9

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:                1936,                 6336,                  5184;
  1:               85184,               418176,                684288,                373248;
  2:             3748096,             24532992,              60217344,              65691648,               26873856;
  3:           164916224,           1349314560,            4415938560,            7226081280,             5912248320,             1934917632;
  4:          7256313856,          71243808768,          291451944960,          635895152640,           780416778240,           510818254848,            139314069504;
  5:        319277809664,        3657182183424,        17953439809536,        48963926753280,         80122789232640,         78666011246592,          42908733407232,          10030613004288;
  6:      14048223625216,      183904018366464,      1053268468826112,      3447060443430912,       7050805452472320,       9230145319600128,        7551937079672832,        3530775777509376,         722204136308736;
  7:     618121839509504,     9103248909139968,     59584901950734336,    227505989266440192,     558423791835807744,     913784386640412672,      996855694516813824,      699093603946856448,      285992837978259456,      51998697814228992;
  8:   27197360938418176,   445047724446842880,   3277169607290388480,  14300376468176240640,   40951078067959234560,   80413026024356315136,   109654126396849520640,   102533728578872279040,    62918424355217080320,   22879427038260756480,    3743906242624487424;
  9: 1196683881290399744, 21540309863227195392, 176238898880949780480, 865172776324662558720, 2831474540698895646720, 6486650765964742754304, 10614519435215033597952, 12406581158043545763840, 10150839129308355624960, 5536821343259103068160, 1812050621430251913216, 269561249468963094528;

%F Formula

The row polynomials satisfy P(n,x) = (44+72*x)*P(n-1,x) for n >= 1, with P(0,x) = 1936 + 6336*x + 5184*x^2.

%t Mathematica

P[0, x_] := 1936 + 6336*x + 5184*x^2;
P[n_, x_] := P[n, x] = Expand[(44 + 72*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] = 1936 + 6336*x + 5184*x^2; for(n = 1, N, my(i = n + 1); v[i] = (44 + 72*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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^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
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool 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.