A200536 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 21\,x + 6\,x^{2}\right)\,P_{n-1}(x)\)

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

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	18	21	6																											45
2	324	756	657	252	36																									2025
3	5832	20412	29646	22869	9882	2268	216																							91125
4	104976	489888	997272	1156680	836001	385560	110808	18144	1296																					4100625
5	1889568	11022480	28868400	44702280	45321930	31436181	15107310	4966920	1069200	136080	7776																			184528125
6	34012224	238085568	762440688	1477012320	1927753020	1785825468	1204022961	595275156	214194780	54704160	9412848	979776	46656																	8303765625
7	612220032	4999796928	18927802656	44025989616	70291457208	81489745764	70741266246	46714387797	23580422082	9054416196	2603387304	543530736	77892192	6858432	279936															373669453125
8	11019960576	102852965376	449369503488	1219950450432	2303358827616	3207091962816	3406376196720	2815364046096	1829897338689	938454682032	378486244080	118781183808	28436528736	5020372224	616419072	47029248	1679616													16815125390625
9	198359290368	2082772548864	10314683099136	32012985473280	69775635377088	113417893413216	142483855725792	141453004737744	112499054244738	72212212665621	37499684748246	15717000526416	5277179841696	1400220906336	287142532416	43913560320	4716361728	317447424	10077696											756680642578125
10	3570467226624	41655450977280	230592675052800	805338718894080	1990122230321280	3698888337196416	5365138977004320	6218822416000320	5850399210252660	4511017995547140	2866445116914897	1503672665182380	650044356694740	230326756148160	66236283666720	15221762704512	2729934472320	368239011840	35145964800	2116316160	60466176									34050628916015625
11	64268410079232	824777929350144	5046855424833024	19588475822065920	54117869292875520	113204589219646848	186189890049130176	246800052028274976	268093290382580520	241369641831156300	181429785272474046	114327563401778517	60476595090824682	26818849092350700	9929381125280760	3046914222571296	766213539296832	155287502358912	24745253448960	2985593022720	256406819328	13967686656	362797056							1532278301220703125
12	1156831381426176	16195639339966464	108549344623822848	463525196294780928	1415760772084141824	3291688716036424704	6053421610254180096	9031616162858564352	11125619659775004912	11455412963164654176	9943058355654298248	7316139482940906072	4578036254707037409	2438713160980302024	1104784261739366472	424274554191283488	137353329133024752	37167144703121664	8303733347399424	1505116010990592	215784296918784	23549519702016	1838292682752	91424858112	2176782336					68952523554931640625
13	20822964865671168	315814967129346048	2300937617656664064	10720163606446135296	35869019087447889408	91762524280191308544	186581616653845011456	309441077043010488576	426245622957505020384	494025143169390264432	486292440586885135632	409227213940664497560	295701931860411490362	183932435144138658453	98567310620137163454	45469690437851610840	18010831132847597616	6099075841597410672	1754097213816893088	424473356711948544	85313953659737088	13986057655874304	1822334963544576	181546911995904	12988860198912	594261577728	13060694016			3102863559971923828125
14	374813367582081024	6121951670507323392	48173929216730247168	243177524689596456960	884571405015370834944	2469295819518526043136	5500696224177915022080	10038728482186081886208	15290173531061380695744	19700257121414693119296	21685265674866158116752	20542381962272890390944	16834160909762672089068	11975887685307124135116	7411004320351849711857	3991962561769041378372	1870462323306963565452	760828961565662607072	267719329319335285392	81071016960554292672	20974174939727545536	4590182204931907584	838392962075585280	125453224585608192	14980294416761856	1372744244551680	90647746818048	3839844040704	78364164096	139628860198736572265625

Technical details

Node id: n-e2e944875f74226f · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-10e88990a58171dc

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:           18,            21,              6;
  2:          324,           756,            657,            252,             36;
  3:         5832,         20412,          29646,          22869,           9882,            2268,             216;
  4:       104976,        489888,         997272,        1156680,         836001,          385560,          110808,           18144,            1296;
  5:      1889568,      11022480,       28868400,       44702280,       45321930,        31436181,        15107310,         4966920,         1069200,         136080,           7776;
  6:     34012224,     238085568,      762440688,     1477012320,     1927753020,      1785825468,      1204022961,       595275156,       214194780,       54704160,        9412848,         979776,         46656;
  7:    612220032,    4999796928,    18927802656,    44025989616,    70291457208,     81489745764,     70741266246,     46714387797,     23580422082,     9054416196,     2603387304,      543530736,      77892192,       6858432,       279936;
  8:  11019960576,  102852965376,   449369503488,  1219950450432,  2303358827616,   3207091962816,   3406376196720,   2815364046096,   1829897338689,   938454682032,   378486244080,   118781183808,   28436528736,    5020372224,    616419072,    47029248,    1679616;
  9: 198359290368, 2082772548864, 10314683099136, 32012985473280, 69775635377088, 113417893413216, 142483855725792, 141453004737744, 112499054244738, 72212212665621, 37499684748246, 15717000526416, 5277179841696, 1400220906336, 287142532416, 43913560320, 4716361728, 317447424, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (18+21*x+6*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[(18 + 21*x + 6*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] = (18 + 21*x + 6*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 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)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

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) 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

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.