A272867 → basis A013611 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(46 + 56\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)

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

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	46	56	16																											118
2	2116	5152	4608	1792	256																									13924
3	97336	355488	534336	422912	185856	43008	4096																							1643032
4	4477456	21803264	46044160	55064576	40781824	19152896	5570560	917504	65536																					193877776
5	205962976	1253687680	3410653440	5460295680	5696286720	4045848576	1981317120	660602880	143523840	18350080	1048576																			22877577568
6	9474296896	69203559936	230391975936	462229196800	622376202240	592465821696	408848695296	206075068416	75296931840	19451084800	3372220416	352321536	16777216																	2699554153024
7	435817657216	3713924383232	14625018999808	35271750664192	58200411938816	69502162272256	61943145234432	41854433230848	21545441820672	8408579178496	2449133797376	516268490752	74457284608	6576668672	268435456															318547390056832
8	20047612231936	195246310432768	887703721967616	2500924384673792	4886437290377216	7020670543724544	7672712359051264	6506154658103296	4326028908429312	2263010315862016	928267657805824	295437110083584	71522211332096	12732430548992	1571958030336	120259084288	4294967296													37588592026706176
9	922190162669056	10103996564895744	52088926390456320	167877871092105216	379031140450566144	636606123427233792	824285315610968064	841285735079215104	686105388386353152	450454567931346944	238645352482209792	101781260954763264	34686782410850304	9317929401188352	1929681561452544	297280456359936	32083405701120	2164663517184	68719476736											4435453859151328768
10	42420747482776576	516426491094671360	2976669464197857280	10801025893140725760	27670016064131235840	53195671480358141952	79631565677238681600	95044819462693847040	91861414079983779840	72603383635745177600	47180828232518729728	25253350829824409600	11113668244081213440	3999584742738493440	1165557916866969600	270823938852913152	48998447201648640	6652732542812160	637716744110080	38482906972160	1099511627776									523383555379856794624
11	1951354384207722496	26131180449390370816	166525410814127456256	671803504937068134400	1925304900393083207680	4169338201978075348992	7084729881079135076352	9682560116895013404672	10822239988425928212480	10004711947126471393280	7705890207577331990528	4965434657364894547968	2680309637418202431488	1210399933111709204480	455411101674535649280	141722500415600197632	36068995816910487552	7383121761908293632	1185863147852922880	143926072075878400	12409088231079936	677299162710016	17592186044416							61759259534823101765632
12	89762301673555234816	1311310146187589517312	9154736672763047313408	40646483119886517665792	128849428267583682379776	310355487791997215834112	590185392246701764378624	908852049949251386474496	1153402084110979609657344	1221183150789989878464512	1087890658402454520987648	820015237017139282968576	524652827454908718383104	285222691136396272336896	131616261129975594221568	51388655642678952722432	16882211457644063883264	4627047373444371972096	1045108456538666696704	191158883785786785792	27604488460525633536	3028881855639191552	237283405367083008	11821949021847552	281474976710656					7287592625109126008344576
13	4129065876983540801536	65346955618348210946048	495987451960382073143296	2403384439528511894454272	8349752541786703135768576	22142264151334742424289280	46590026211981464166203392	79823264068152818036375552	113395176942210367916867584	135306574445742414733770752	136883660076528014913961984	118181508185965698247294976	87461133870324873227665408	55621045922023345284448256	30421263954895608079187968	14297951841024205459030016	5760206621121257798631424	1980466511089371931738112	577305025235968520617984	141352140195421478715392	28696439265806906294272	4743722059721426862080	622204235931090681856	62293790045788700672	4471511480025481216	204913783045357568	4503599627370496			859935929762876868984659968
14	189937030341242876870656	3237187647555095988404224	26540917358836811830394880	139376532817986514616057856	526613944767151123505086464	1524584444333909717431615488	3516704038894480177577852928	6635187841427347501769293824	10431721346549938160872194048	13851404558358376769680048128	15688139363557229797803294720	15266739531971869590297772032	12831515177673471508944519168	9347295740126717955790602240	5913534855483703279214395392	3251233300913641028101079040	1552395030947263093709733888	642440259749288832927793152	229625461719799504903864320	70518744039633576083324928	18472636460946216827486208	4086846056759116047581184	753412858450140576350208	113608504509790188208128	13649409545542790283264	1256531317634132606976	83226521113806766080	3530822107858468864	72057594037927936	101472439712019470540189876224

Technical details

Node id: n-606569787e0702d3 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-915220761d896620

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:               1;
  1:              46,                56,                16;
  2:            2116,              5152,              4608,               1792,                256;
  3:           97336,            355488,            534336,             422912,             185856,              43008,               4096;
  4:         4477456,          21803264,          46044160,           55064576,           40781824,           19152896,            5570560,             917504,              65536;
  5:       205962976,        1253687680,        3410653440,         5460295680,         5696286720,         4045848576,         1981317120,          660602880,          143523840,           18350080,            1048576;
  6:      9474296896,       69203559936,      230391975936,       462229196800,       622376202240,       592465821696,       408848695296,       206075068416,        75296931840,        19451084800,         3372220416,          352321536,          16777216;
  7:    435817657216,     3713924383232,    14625018999808,     35271750664192,     58200411938816,     69502162272256,     61943145234432,     41854433230848,     21545441820672,      8408579178496,      2449133797376,       516268490752,       74457284608,       6576668672,        268435456;
  8:  20047612231936,   195246310432768,   887703721967616,   2500924384673792,   4886437290377216,   7020670543724544,   7672712359051264,   6506154658103296,   4326028908429312,   2263010315862016,    928267657805824,    295437110083584,    71522211332096,   12732430548992,    1571958030336,    120259084288,     4294967296;
  9: 922190162669056, 10103996564895744, 52088926390456320, 167877871092105216, 379031140450566144, 636606123427233792, 824285315610968064, 841285735079215104, 686105388386353152, 450454567931346944, 238645352482209792, 101781260954763264, 34686782410850304, 9317929401188352, 1929681561452544, 297280456359936, 32083405701120, 2164663517184, 68719476736;

%F Formula

The row polynomials satisfy P(n,x) = (46+56*x+16*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[(46 + 56*x + 16*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] = (46 + 56*x + 16*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_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    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 Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - ↑ Geometric basis map (elementary) (above)

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.