A272866 → basis A013609 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(19 + 36\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)

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

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	19	36	16																											71
2	361	1368	1904	1152	256																									5041
3	6859	38988	91200	112320	76800	27648	4096																							357911
4	130321	987696	3246112	6041088	6961920	5087232	2301952	589824	65536																					25411681
5	2476099	23457780	99318320	247443840	401693440	443943936	338268160	175472640	59310080	11796480	1048576																			1804229351
6	47045881	534837384	2771145744	8652216960	18129246720	26855000064	28836171776	22614736896	12856197120	5166858240	1393557504	226492416	16777216																	128100283921
7	893871739	11855562012	72658649056	272710767168	700273830144	1301333354496	1804735213568	1897462185984	1519777021952	922829193216	418183643136	137141157888	30769414144	4227858432	268435456															9095120158391
8	16983563041	257435060832	1821616512320	7986904934400	24285328775680	54298563895296	92342351101952	121843582894080	126060165529600	102605122437120	65483772526592	32425560244224	12212638842880	3382286745600	649613803520	77309411328	4294967296													645753531245761
9	322687697779	5502674425284	44150112932688	221448349170432	778095688573440	2033735028885504	4097818231578624	6508129736982528	8258989746880512	8437160611676160	6954938734215168	4615183414591488	2447100667232256	1022727410417664	329507206594560	78971563671552	13258564042752	1391569403904	68719476736											45848500718449031
10	6131066257801	116167571200440	1042111428195760	5884965490619520	23462360459953920	70195583924195328	163522538457047040	303715681801666560	456778567427358720	561759758301265920	568025453920518144	473060849095802880	323920535350149120	181370379451760640	82232434376048640	29726357713846272	8367004314501120	1767293142958080	263539193282560	24739011624960	1099511627776									3255243551009881201
11	116490258898219	2427902238089196	24080246759060096	151189036876025280	674317389252559360	2272520518967964672	6009367019314188288	12780538681472483328	22228917941292564480	31976914743935631360	38324292002173157376	38425228606779162624	32273088001830027264	22676150067721666560	13274478479010693120	6427094505329000448	2544840909690765312	810412448459784192	202501866850877440	38234142466375680	5128122231947264	435406604599296	17592186044416							231122292121701565271
12	2213314919066161	50323791844030608	546793013135724384	3778327019780070912	18640119671480499456	69872340463499870208	206777790277857255424	495527775946775396352	978598705726595137536	1612291044924869050368	2234993165883653750784	2621384491510007857152	2609685573913590890496	2207481677061059248128	1584925901568463601664	962814421929182625792	492119035140139646976	209845621354867458048	73739838169771147264	20983115088849272832	4713893320810561536	804631405299499008	98058845011574784	7599824371187712	281474976710656					16409682740640811134241
13	42052983462257059	1035831382122963348	12236136794668923760	92277942518211914880	498930734680383632640	2059072009296276648960	6742424186708971171840	17976985640407591649280	39740819987334937968640	73791527674878343249920	116164927060710229278720	156062716029299589120000	179713758252856969789440	177832984389209523486720	151337901686616395612160	110670513306096107520000	69370395282208645447680	37108382821654431006720	16829403856243114639360	6410843302477776814080	2024793527010312847360	520717997671459061760	106252141778970869760	16548617568261242880	1847883222105456640	131730289100587008	4503599627370496			1165087474585497590531111
14	799006685782884121	21194703664977557736	270449376590532344912	2210355134568075051648	12997468078297620736000	58560321705414457743360	210215543637022549749760	617235150038007629905920	1510125849801380659527680	3120340315613267755008000	5499481730246473720791040	8327793421540313939312640	10891458016830431302778880	12345525456966625284587520	12152827702102965878784000	10396231963761368660705280	7723582416367286464020480	4973121716668483145564160	2765586779363517021552640	1321398686718300782592000	538531897602285743636480	185359701767773413703680	53161335042137978306560	12470988800583259914240	2330894282140942336000	333804552580887478272	34393990354228477952	2269814212194729984	72057594037927936	82721210695570328927708881

Technical details

Node id: n-414aa3fe097fc93d · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-67cad0a0ca30ceab

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:           19,            36,             16;
  2:          361,          1368,           1904,            1152,             256;
  3:         6859,         38988,          91200,          112320,           76800,            27648,             4096;
  4:       130321,        987696,        3246112,         6041088,         6961920,          5087232,          2301952,           589824,            65536;
  5:      2476099,      23457780,       99318320,       247443840,       401693440,        443943936,        338268160,        175472640,         59310080,         11796480,          1048576;
  6:     47045881,     534837384,     2771145744,      8652216960,     18129246720,      26855000064,      28836171776,      22614736896,      12856197120,       5166858240,       1393557504,        226492416,         16777216;
  7:    893871739,   11855562012,    72658649056,    272710767168,    700273830144,    1301333354496,    1804735213568,    1897462185984,    1519777021952,     922829193216,     418183643136,     137141157888,      30769414144,       4227858432,       268435456;
  8:  16983563041,  257435060832,  1821616512320,   7986904934400,  24285328775680,   54298563895296,   92342351101952,  121843582894080,  126060165529600,  102605122437120,   65483772526592,   32425560244224,   12212638842880,    3382286745600,    649613803520,    77309411328,     4294967296;
  9: 322687697779, 5502674425284, 44150112932688, 221448349170432, 778095688573440, 2033735028885504, 4097818231578624, 6508129736982528, 8258989746880512, 8437160611676160, 6954938734215168, 4615183414591488, 2447100667232256, 1022727410417664, 329507206594560, 78971563671552, 13258564042752, 1391569403904, 68719476736;

%F Formula

The row polynomials satisfy P(n,x) = (19+36*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[(19 + 36*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] = (19 + 36*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 (−∞, 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 read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) 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
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (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.