A272866 → basis A013610 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(29 + 33\,x + 9\,x^{2}\right)\,P_{n-1}(x)\)

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

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	29	33	9																											71
2	841	1914	1611	594	81																									5041
3	24389	83259	117450	87615	36450	8019	729																							357911
4	707281	3219348	6373098	7166016	5005395	2223936	613818	96228	6561																					25411681
5	20511149	116701365	297423855	447100830	438992865	294166323	136239165	43062030	8890155	1082565	59049																			1804229351
6	594823321	4061207502	12661037181	23831223570	30161935170	27041495382	17609360229	8392188222	2905013970	712327770	117448461	11691702	531441																	128100283921
7	17249876309	137404187151	506543335704	1145470578021	1775075832369	1994028238818	1674498210777	1067855804433	519671858517	192052660338	53057947509	10625808501	1458274104	122762871	4782969															9095120158391
8	500246412961	4553967345576	19379343798180	51171214525200	93836618234730	126713556596088	130339062484848	104172513433560	65380209340275	32329400720760	12553464995568	3787534657368	870463157130	147315445200	17314347780	1262703816	43046721													645753531245761
9	14507145975869	148573184649417	716784110267877	2164469272680924	4584326102322390	7231842473759442	8805909743844066	8462613960938016	6506770576539087	4032652150033155	2019342592719027	815067456404256	263213260210026	67085244012402	13197694191390	1933830894204	198746710857	12784876137	387420489											45848500718449031
10	420707233300201	4787358172036770	25820218604982015	87760643208431490	210823999958230695	380486416569791004	535281119126441010	601097408677891620	547215795125184645	407833867025193510	250199391328797681	126569131145749710	52704493941902445	17967116770928820	4965465525849810	1095372200359404	188359321846095	24333880914090	2221856504415	127848761370	3486784401									3255243551009881201
11	12200509765705829	152716725687972963	910555524321393654	3440212090557250635	9242379192111767460	18781143868021435461	29976620201093968677	38520479531959529346	40523002617138747255	35295180060962729655	25639242116493180384	15597589520303806653	7957006174084090464	3399416866751463855	1211253799167417555	357330207101825826	86298892678045377	16779889970664381	2562685805407860	296033806952235	24316834412574	1265702737563	31381059609							231122292121701565271
12	353814783205469041	4831401867219508284	31555566740914876206	131188933459958015664	389750995278523070181	880613594327325210264	1572281146205438368986	2275352667875120236524	2716132482261533856906	2707503623921133583524	2272985986944321035046	1616081706481749912504	976226812266902866461	501542598563301696984	218920172345410230486	80928703179240902964	25195848914530326186	6550452131441196924	1404744291491160906	244172621716053624	33538476076059141	3503465177574384	261529750781406	12426899605164	282429536481					16409682740640811134241
13	10260628712958602189	151786541995146218589	1077732030153624404715	4889295389593948943610	15916013767924017438015	39580277480823314554605	78164160810266152270935	125796027542103879928110	168005010341312394976440	188628151019219573328810	179709405551136522984780	146242439672423543870850	102098147752136819555415	61304955521479293018285	31685632061007978482715	14085181466666239064850	5371608374545021331580	1749784454604463601610	483665145021576100440	112391636831944637310	21673010946255932835	3405923796104091405	425044998317147715	40522048462505610	2772045900561015	121162271150349	2541865828329			1165087474585497590531111
14	297558232675799463481	4740410465386874211318	36355530418711560369873	178720802171250440687586	632610735397779440484000	1717060159791714238030530	3716153944276203953101215	6583724602787205373847490	9726891656079882862382805	12146545968699611580111000	12948346837689016641197310	11869094492860764190711680	9404231443962173421708105	6463264542995226437696610	3860830191747279421600875	2005840720239897859974810	905758319810863314100305	354773458743511299972480	120113651578478197750110	34968367345278578211000	8690427638089779987405	1825505882800966170090	319778888738935695615	45854938514815359930	5243021915233284000	459689656744424106	29020482162032193	1174342012687998	22876792454961	82721210695570328927708881

Technical details

Node id: n-6a3923af89150a19 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-40d39f31cccb6263

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:             29,              33,               9;
  2:            841,            1914,            1611,              594,               81;
  3:          24389,           83259,          117450,            87615,            36450,             8019,              729;
  4:         707281,         3219348,         6373098,          7166016,          5005395,          2223936,           613818,            96228,             6561;
  5:       20511149,       116701365,       297423855,        447100830,        438992865,        294166323,        136239165,         43062030,          8890155,          1082565,            59049;
  6:      594823321,      4061207502,     12661037181,      23831223570,      30161935170,      27041495382,      17609360229,       8392188222,       2905013970,        712327770,        117448461,        11691702,          531441;
  7:    17249876309,    137404187151,    506543335704,    1145470578021,    1775075832369,    1994028238818,    1674498210777,    1067855804433,     519671858517,     192052660338,      53057947509,     10625808501,      1458274104,      122762871,        4782969;
  8:   500246412961,   4553967345576,  19379343798180,   51171214525200,   93836618234730,  126713556596088,  130339062484848,  104172513433560,   65380209340275,   32329400720760,   12553464995568,   3787534657368,    870463157130,   147315445200,    17314347780,    1262703816,     43046721;
  9: 14507145975869, 148573184649417, 716784110267877, 2164469272680924, 4584326102322390, 7231842473759442, 8805909743844066, 8462613960938016, 6506770576539087, 4032652150033155, 2019342592719027, 815067456404256, 263213260210026, 67085244012402, 13197694191390, 1933830894204, 198746710857, 12784876137, 387420489;

%F Formula

The row polynomials satisfy P(n,x) = (29+33*x+9*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[(29 + 33*x + 9*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] = (29 + 33*x + 9*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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (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, 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)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013610 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) (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.