A200536 → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(45 + 57\,x + 18\,x^{2}\right)\,P_{n-1}(x)\)

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

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	45	57	18																											120
2	2025	5130	4869	2052	324																									14400
3	91125	346275	547965	462213	219186	55404	5832																							1728000
4	4100625	20776500	46036350	58266540	46072881	23306616	7365816	1329696	104976																					207360000
5	184528125	1168678125	3329707500	5620043250	6223126725	4723749657	2489250690	899206920	213101280	29918160	1889568																			24883200000
6	8303765625	63108618750	219772996875	463731480000	660317902875	668447736390	493286292549	267379094556	105650864460	29678814720	5626188720	646232256	34012224																	2985984000000
7	373669453125	3313202484375	13636443909375	34530932559375	60102913933125	76065435241425	72185126390685	52181437185333	28874050556274	12170469638628	3846586491720	883991873520	139637185632	13570877376	612220032															358318080000000
8	16815125390625	170393270625000	809218567687500	2390806912725000	4918350273243750	7470367466121000	8665912947138300	7831894711954680	5573006469628641	3132757884781872	1386546071542128	478103517831744	125909766995040	24481862786304	3314559253248	279172334592	11019960576													42998169600000000
9	756680642578125	8626159325390625	46429924228593750	156778848302062500	372167690539668750	659547025979388750	904307333108508000	980858914415021700	853189722763195005	599609578399201017	341275889105278002	156937426306403472	57875669318944512	16884403865072352	3810997151126208	642166162645248	76070787856128	5653239775488	198359290368											5159780352000000000
10	34050628916015625	431307966269531250	2594657923400390625	9856824722479687500	26519679063617343750	53715193799270737500	84987028900422056250	107556015603489930000	110580027641953156125	93269705684936551650	64892575988229478149	37307882273974620660	17692804422712504980	6883584998623355520	2175667939850804640	550043584504532352	108624605444576640	16149421625310720	1700435016679680	113064795509760	3570467226624									619173642240000000000
11	1532278301220703125	21349744330341796875	141957071950869140625	599216157538259765625	1801928409665329687500	4106228272598006156250	7363536570222536756250	10651154837867977331250	12636560653494415048125	12436206612276293463375	10226979641066866771005	7056586235986796113893	4090791856426746708402	1989793057964206954140	808739881823642563080	272669563849420219680	75402614479078776384	16819111004561433216	2952279506395676160	392702301004273920	37213194669488640	2238682951093248	64268410079232							74300837068800000000000
12	68952523554931640625	1048078358034960937500	7632584674040566406250	35440575588367382812500	117797326709736287109375	298276082453522745000000	597848868572076439312500	972935661113507685825000	1308304713429729046659375	1471634041883238455557500	1396535952510657202973850	1124335939181190513413220	770396682529654582248081	449734375672476205365288	223445752401705152475816	94184578680527261155680	33492600663801063594480	9962861169802318702848	2448788965671225095424	488695533491851665408	77199656032492773120	9290534247036979200	800334510716676096	43959592494194688	1156831381426176					8916100448256000000000000
13	3102863559971923828125	51093819954204345703125	404447922163807031250000	2048748638341473808593750	7458379034607803935546875	20774801693454104780859375	46025287666369489402031250	83228459742879612312937500	125092324422105121099321875	158309742450283424504521875	170276743092059288940469500	156687079310159325873139350	123892646392354145119246485	84388704714713169671016537	49557058556941658047698594	25069932689625492139702296	10897711557891794492190048	4052729406727255667315760	1280945402082356440057056	340903771106834892033792	75407831312579771436288	13614974037822082109184	1955169313648228154880	214826865219754924032	16963775377233444864	857212053636796416	20822964865671168			1069932053790720000000000000
14	139628860198736572265625	2476085120857595214843750	21168355778840458740234375	116166909047879000390625000	459685791541763710751953125	1396871156668226068019531250	3389552464136451466440234375	6742668515894817336053437500	11201631982333519160543484375	15752313177694579026997706250	18937770598406715378866668125	19606268289309650774979426000	17571313983992085306063485775	13679739984109146772709302110	9270291438863399895540816069	5471895993643658709083720844	2811410237438733648970157724	1254801170515817649598683264	484806927319211913698986704	161303686939592489236456512	45881884599638094481586112	11047188096442068723389952	2221377102896464833046272	366181392493635454374912	48201548855169642476544	4872393312871550828544	355146077266454605824	16616725962805592064	374813367582081024	128391846454886400000000000000

Technical details

Node id: n-d3d706d2f2257431 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-d7967bc76bb8e021

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:              45,               57,                18;
  2:            2025,             5130,              4869,               2052,                324;
  3:           91125,           346275,            547965,             462213,             219186,              55404,               5832;
  4:         4100625,         20776500,          46036350,           58266540,           46072881,           23306616,            7365816,            1329696,             104976;
  5:       184528125,       1168678125,        3329707500,         5620043250,         6223126725,         4723749657,         2489250690,          899206920,          213101280,           29918160,            1889568;
  6:      8303765625,      63108618750,      219772996875,       463731480000,       660317902875,       668447736390,       493286292549,       267379094556,       105650864460,        29678814720,         5626188720,          646232256,          34012224;
  7:    373669453125,    3313202484375,    13636443909375,     34530932559375,     60102913933125,     76065435241425,     72185126390685,     52181437185333,     28874050556274,     12170469638628,      3846586491720,       883991873520,      139637185632,       13570877376,        612220032;
  8:  16815125390625,  170393270625000,   809218567687500,   2390806912725000,   4918350273243750,   7470367466121000,   8665912947138300,   7831894711954680,   5573006469628641,   3132757884781872,   1386546071542128,    478103517831744,   125909766995040,    24481862786304,    3314559253248,    279172334592,    11019960576;
  9: 756680642578125, 8626159325390625, 46429924228593750, 156778848302062500, 372167690539668750, 659547025979388750, 904307333108508000, 980858914415021700, 853189722763195005, 599609578399201017, 341275889105278002, 156937426306403472, 57875669318944512, 16884403865072352, 3810997151126208, 642166162645248, 76070787856128, 5653239775488, 198359290368;

%F Formula

The row polynomials satisfy P(n,x) = (45+57*x+18*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[(45 + 57*x + 18*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] = (45 + 57*x + 18*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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    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
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

A200536 → basis A007318 → x ↦ x/(x+1)

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.