A139548 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(32 + 96\,x + 72\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	32	96	72																											200
2	1024	6144	13824	13824	5184																									40000
3	32768	294912	1105920	2211840	2488320	1492992	373248																							8000000
4	1048576	12582912	66060288	198180864	371589120	445906944	334430208	143327232	26873856																					1600000000
5	33554432	503316480	3397386240	13589544960	35672555520	64210599936	80263249920	68797071360	38698352640	12899450880	1934917632																			320000000000
6	1073741824	19327352832	159450660864	797253304320	2690729902080	6457751764992	11301065588736	14529941471232	13621820129280	9081213419520	4086546038784	1114512556032	139314069504																	64000000000000
7	34359738368	721554505728	7035156430848	42210938585088	174120121663488	522360364990464	1175310821228544	2014818550677504	2644449347764224	2644449347764224	1983337010823168	1081820187721728	405682570395648	93619054706688	10030613004288															12800000000000000
8	1099511627776	26388279066624	296868139499520	2078076976496640	10130625260421120	36470250937516032	100293190078169088	214913978738933760	362667339121950720	483556452162600960	507734274770731008	415418952085143552	259636845053214720	119832390024560640	38517553936465920	7703510787293184	722204136308736													2560000000000000000
9	35184372088832	949978046398464	12112220091580416	96897760732643328	545049904121118720	2289209597308698624	7439931191253270528	19131251634651267072	39458206496468238336	65763677494113730560	88780964617053536256	96851961400422039552	84745466225369284608	58669938156024889344	31430324012156190720	12572129604862476288	3535911451367571456	623984373770747904	51998697814228992											512000000000000000000
10	1125899906842624	33776997205278720	481322210175221760	4331899891576995840	27615861808803348480	132556136682256072704	497085512558460272640	1491256537675380817920	3634937810583740743680	7269875621167481487360	11995294774926344454144	16357220147626833346560	18401872666080187514880	16986343999458634629120	12739757999593975971840	7643854799756385583104	3583056937385805742080	1264608330842049085440	316152082710512271360	49918749901659832320	3743906242624487424									102400000000000000000000
11	36028797018963968	1188950301625810944	18725967250606522368	187259672506065223680	1334225166605714718720	7204815899670859481088	30620467573601152794624	104984460252346809581568	295268794459725401948160	688960520406025937879040	1343473014791750578864128	2198410387841046401777664	3022814283281438802444288	3487862634555506310512640	3363296111892809656565760	2690636889514247725252608	1765730458743725069697024	934798478158442683957248	389499365899351118315520	122999799757689826836480	27674954945480211038208	3953564992211458719744	269561249468963094528							20480000000000000000000000
12	1152921504606846976	41505174165846491136	715964254360851972096	7875606797969371693056	62020403534008802082816	372122421204052812496896	1767581500719250859360256	6817814359917110457532416	21731783272235789583384576	57951422059295438889025536	130390699633414737500307456	248927699300155407955132416	404507511362752537927090176	560087323425349667899047936	660102916894162108595306496	660102916894162108595306496	556961836129449279127289856	393149531385493608795734016	229337226641537938464178176	108633423145991655061979136	40737533679746870648242176	11639295337070534470926336	2380764955309882050871296	310534559388245484896256	19408409961765342806016					4096000000000000000000000000
13	36893488147419103232	1438846037749345026048	26978363207800219238400	323740358493602630860800	2792260592007322691174400	18428919907248329761751040	96751829513053731249192960	414650697913087419639398400	1477193111315373932465356800	4431579333946121797396070400	11300527301562610583359979520	24655695930682059454603591680	46229429870028861477381734400	74678309790046622386539724800	104016217207564938324108902400	124819460649077925988930682880	128720068794361611176084766720	113576531289142598096545382400	85182398466856948572409036800	53799409558014914887837286400	28244690017957830316114575360	12104867150553355849763389440	4126659255870462221510246400	1076519805879251014307020800	201847463602359565182566400	24221695632283147821907968	1397405517247104682033152			819200000000000000000000000000
14	1180591620717411303424	49584848070131274743808	1004093173420158313562112	13053211254462058076307456	122373855510581794465382400	881091759676188920150753280	5066277618138086290866831360	23883880199793835371229347840	94042778286688226774215557120	313475927622294089247385190400	893406393723538154355047792640	2192906602775957287962390036480	4659926530898909236920078827520	8602941287813370898929376296960	13826155641128631801850783334400	19356617897580084522591096668160	23590878062675728011907899064320	24978576772244888483196599009280	22897028707891147776263549091840	18076601611493011402313328230400	12201706087757782696561496555520	6972403478718732969463712317440	3327738023933940735425862696960	1302158357191542026905772359680	406924486622356883408053862400	97661876789365652017932926976	16903017136620978233873006592	1878113015180108692652556288	100613197241791537106386944	163840000000000000000000000000000

Technical details

Node id: n-08ea8d2f5c08b286 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-6b852c5cd7d4c499

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
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:              1;
  1:             32,              96,                72;
  2:           1024,            6144,             13824,             13824,               5184;
  3:          32768,          294912,           1105920,           2211840,            2488320,             1492992,              373248;
  4:        1048576,        12582912,          66060288,         198180864,          371589120,           445906944,           334430208,            143327232,             26873856;
  5:       33554432,       503316480,        3397386240,       13589544960,        35672555520,         64210599936,         80263249920,          68797071360,          38698352640,          12899450880,           1934917632;
  6:     1073741824,     19327352832,      159450660864,      797253304320,      2690729902080,       6457751764992,      11301065588736,       14529941471232,       13621820129280,        9081213419520,        4086546038784,        1114512556032,         139314069504;
  7:    34359738368,    721554505728,     7035156430848,    42210938585088,    174120121663488,     522360364990464,    1175310821228544,     2014818550677504,     2644449347764224,     2644449347764224,     1983337010823168,     1081820187721728,      405682570395648,       93619054706688,       10030613004288;
  8:  1099511627776,  26388279066624,   296868139499520,  2078076976496640,  10130625260421120,   36470250937516032,  100293190078169088,   214913978738933760,   362667339121950720,   483556452162600960,   507734274770731008,   415418952085143552,   259636845053214720,   119832390024560640,    38517553936465920,     7703510787293184,     722204136308736;
  9: 35184372088832, 949978046398464, 12112220091580416, 96897760732643328, 545049904121118720, 2289209597308698624, 7439931191253270528, 19131251634651267072, 39458206496468238336, 65763677494113730560, 88780964617053536256, 96851961400422039552, 84745466225369284608, 58669938156024889344, 31430324012156190720, 12572129604862476288, 3535911451367571456, 623984373770747904, 51998697814228992;

%F Formula

The row polynomials satisfy P(n,x) = (32+96*x+72*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(32 + 96*x + 72*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (32 + 96*x + 72*x^2)*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_A013609 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
    
    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
    
    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
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from basis_map_A013609 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
    
    ↑ Geometric basis map (elementary) (above)

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.