Veronese parent of A098172 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(2 + 20\,x + 64\,x^{2} + 64\,x^{3}\right)\,P_{n-1}(x)\)

\(P_{1}(x) = 1 + 8\,x + 16\,x^{2}\)

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	x²⁹	x³⁰	x³¹	x³²	x³³	x³⁴	x³⁵	x³⁶	x³⁷	x³⁸	x³⁹	x⁴⁰	x⁴¹	x⁴²	x⁴³	x⁴⁴	P_n(1)
1	1	8	16																																											25
2	2	36	256	896	1536	1024																																								3750
3	4	112	1360	9344	39680	106496	176128	163840	65536																																					562500
4	8	304	5216	53312	360448	1691648	5619712	13205504	21495808	23068672	14680064	4194304																																		84375000
5	16	768	17024	230912	2140416	14338048	71553024	270139392	775028736	1680867328	2711617536	3154116608	2499805184	1207959552	268435456																															12656250000
6	32	1856	50432	852480	10037760	87352320	581632000	3025960960	12449873920	40730624000	105931341824	217717932032	349200973824	427819008000	386547056640	242665652224	94489280512	17179869184																												1898437500000
7	64	4352	140032	2834432	40471552	433246208	3607285760	23917527040	128233963520	561344675840	2016928595968	5957614567424	14439126401024	28553211019264	45612284248064	57945551273984	57161719742464	42193758715904	21921513078784	7146825580544	1099511627776																									284765625000000
8	128	9984	371200	8752128	146872320	1866289152	18651078656	150298705920	993412513792	5448956641280	24998446104576	96386833973248	313040033546240	856259712516096	1967680210862080	3779646382407680	6019826162073600	7855323385692160	8254583545528320	6811474534072320	4248512929726464	1882363906752512	527765581332480	70368744177664																						42714843750000000
9	256	22528	950272	25575424	493182976	7253917696	84587905024	802461319168	6305910685696	41578949771264	232173543096320	1105054215962624	4502450522292224	15741978020937728	47263874193293312	121748080730046464	268364735068504064	503936071982120960	800781877651701760	1066724191035719680	1175760560139010048	1052962703502475264	746542006980837376	403072166649659392	155655662120992768	38280596832649216	4503599627370496																			6407226562500000000
10	512	50176	2367488	71614464	1560133632	26069139456	347454701568	3792495181824	34538924408832	266047263604736	1750861889929216	9918210363097088	48626144907362304	207115543067492352	768247612053651456	2484417072092676096	7004065626448723968	17191931960428068864	36647505406053056512	67576337586345476096	107187953717557067776	145141525005780123648	166071362159193882624	158375273345971126272	123541056128167182336	76764981548436946944	36533200177229463552	12501992565580496896	2738188573441261568	288230376151711744																961083984375000000000
11	1024	110592	5771264	193822720	4707287040	88075796480	1320724070400	16302357872640	168833278279680	1487829807923200	11275879905755136	74091174557646848	424698682902839296	2133574608475914240	9425644822766551040	36701254415649079296	126119714622504173568	382635716235195580416	1024396542727317094400	2416646628853866823680	5011626650616815681536	9104368075876207689728	14419887867883753046016	19787264428169400156160	23332212357734093291520	23388935757990778109952	19651011117668049616896	13575254399978414538752	7506599858901142732800	3193592567760966123520	981136200420426776576	193690812773950291968	18446744073709551616													144162597656250000000000
12	2048	241664	13819904	510214144	13667467264	283071348736	4716635095040	64957314498560	753876905492480	7480202425794560	64157036683722752	479726384747380736	3148098278634029056	18224614360666341376	93445332692522500096	425644895934869405696	1726054561157507710976	6239787276179074973696	20122049428600353980416	57881571687146859790336	148406250451911779024896	338668032145284499767296	686336627139415340220416	1231395948709567356993536	1947962093679493676793856	2702679865618065226989568	3266727251692975242084352	3410324210559709591109632	3041074887759535550758912	2283000175442918492864512	1415072796324351841402880	704822420940331402919936	271093350907235570548736	75557863725914323419136	13576803638250229989376	1180591620717411303424										21624389648437500000000000
13	4096	524288	32604160	1312423936	38439157760	873030221824	16002068774912	243238615121920	3126881313357824	34497075735429120	330323511962763264	2769574580371587072	20475507555224780800	134219733265603887104	783563731054782054400	4088050054635412520960	19111883652416489390080	80222440407661386137600	302748609634694342901760	1028018009535842091008000	3141601483752644206919680	8637692824736626261360640	21348454514084273559961600	47362278526428003361095680	94124141355771124252672000	167099486459160896611549184	264065966528647161186353152	369796278850022693866045440	456330689494619852944441344	492718791690248377325322240	461379691393938762619682816	370451904813324758425796608	251315305313726400592609280	141246276650536267694473216	63996920280039073230356480	22459574992528032636338176	5728230543720879644213248	944473296573929042739200	75557863725914323419136							3243658447265625000000000000
14	8192	1130496	75956224	3310747648	105247014912	2600925396992	52008843214848	864852646035456	12198541264945152	148123181507411968	1566276214056812544	14553552651120803840	119691024330565615616	876343095545101811712	5739207384099809394688	33747870142899500875776	178722910114045929455616	854465836147825012899840	3694741784673938130862080	14468404951610555073822720	51353710360932896892518400	165276478951440468431339520	482306413093364195414507520	1275568483077854787891363840	3054607282924639904291880960	6614168714626530001224204288	12925252534701256421884821504	22739224068428637946769309696	35903175947209409282275540992	50679235077507764536892981248	63649261190649968067616964608	70707664433336991506510118912	68973971624281924926181474304	58546020816840465894151815168	42746020821702077677528154112	26448798801298648300253085696	13596212564494824231204290560	5649669254911860226438201344	1823060135978860795456913408	428564203053386042433339392	65281994259189975434133504	4835703278458516698824704				486548767089843750000000000000
15	16384	2424832	175046656	8213495808	281642532864	7523540140032	162983891173376	2943077190729728	45189160641757184	598896323625746432	6931073268564623360	70693219840808714240	640174662999657676800	5178175725072883384320	37596929605929585868800	246026071639837904994304	1455798453783356193636352	7810562836304524472221696	38076930228748884060930048	168955725357459613075636224	683224807486669736462254080	2520068556243746439812874240	8482757785218318685301637120	26061597344014888864799784960	73065889318273874052545249280	186824476442703873019861794816	435265128386092773167981395968	922785162674157278284264505344	1777113793219969908578914926592	3101948491699516954313161179136	4893996824932290249531963998208	6955674858261572468792144953344	8869124993077398566671201861632	10095414714054484843056066461696	10196037165667824124930354577408	9069097530268471862652399779840	7038859066021685403054988328960	4711692045671766667654961561600	2679520232780977168132949934080	1269054767567711793283124428800	486956529066592246200822398976	145419268989804514167056498688	31702870693574035477494759424	4487532642409503496509325312	309485009821345068724781056	72982315063476562500000000000000

Technical details

Node id: n-22c2c81430f30e6b · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-5719ac3c88c372d2

Observed on the stored rows ?

Degrees: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44 (affine: slope 3, intercept -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   1,     8,      16;
   2:   2,    36,     256,      896,       1536,        1024;
   3:   4,   112,    1360,     9344,      39680,      106496,       176128,        163840,          65536;
   4:   8,   304,    5216,    53312,     360448,     1691648,      5619712,      13205504,       21495808,        23068672,         14680064,          4194304;
   5:  16,   768,   17024,   230912,    2140416,    14338048,     71553024,     270139392,      775028736,      1680867328,       2711617536,       3154116608,        2499805184,         1207959552,          268435456;
   6:  32,  1856,   50432,   852480,   10037760,    87352320,    581632000,    3025960960,    12449873920,     40730624000,     105931341824,     217717932032,      349200973824,       427819008000,       386547056640,        242665652224,         94489280512,          17179869184;
   7:  64,  4352,  140032,  2834432,   40471552,   433246208,   3607285760,   23917527040,   128233963520,    561344675840,    2016928595968,    5957614567424,    14439126401024,     28553211019264,     45612284248064,      57945551273984,      57161719742464,       42193758715904,       21921513078784,        7146825580544,         1099511627776;
   8: 128,  9984,  371200,  8752128,  146872320,  1866289152,  18651078656,  150298705920,   993412513792,   5448956641280,   24998446104576,   96386833973248,   313040033546240,    856259712516096,   1967680210862080,    3779646382407680,    6019826162073600,     7855323385692160,     8254583545528320,     6811474534072320,      4248512929726464,      1882363906752512,       527765581332480,        70368744177664;
   9: 256, 22528,  950272, 25575424,  493182976,  7253917696,  84587905024,  802461319168,  6305910685696,  41578949771264,  232173543096320, 1105054215962624,  4502450522292224,  15741978020937728,  47263874193293312,  121748080730046464,  268364735068504064,   503936071982120960,   800781877651701760,  1066724191035719680,   1175760560139010048,   1052962703502475264,    746542006980837376,    403072166649659392,    155655662120992768,    38280596832649216,     4503599627370496;
  10: 512, 50176, 2367488, 71614464, 1560133632, 26069139456, 347454701568, 3792495181824, 34538924408832, 266047263604736, 1750861889929216, 9918210363097088, 48626144907362304, 207115543067492352, 768247612053651456, 2484417072092676096, 7004065626448723968, 17191931960428068864, 36647505406053056512, 67576337586345476096, 107187953717557067776, 145141525005780123648, 166071362159193882624, 158375273345971126272, 123541056128167182336, 76764981548436946944, 36533200177229463552, 12501992565580496896, 2738188573441261568, 288230376151711744;

%F Formula

The row polynomials satisfy P(n,x) = (2+20*x+64*x^2+64*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 8*x + 16*x^2.

%t Mathematica

P[1, x_] := 1 + 8*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 20*x + 64*x^2 + 64*x^3)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + 8*x + 16*x^2; for(n = 2, N, my(i = n); v[i] = (2 + 20*x + 64*x^2 + 64*x^3)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
  - 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

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.