A272867 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(33 + 36\,x + 9\,x^{2}\right)\,P_{n-1}(x)\)

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

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	33	36	9																											78
2	1089	2376	1890	648	81																									6084
3	35937	117612	157707	110808	43011	8748	729																							474552
4	1185921	5174928	9761796	10392624	6827814	2834352	726084	104976	6561																					37015056
5	39135393	213465780	519109965	740955600	687308490	432868536	187447770	55112400	10530405	1180980	59049																			2887174368
6	1291467969	8453244888	25167615462	45060685560	54027571455	45696367728	27954820116	12462645744	4018579695	914078520	139237542	12754584	531441																	225199600704
7	42618442977	325449928188	1146471337935	2469115984104	3631603077333	3858518877444	3053826445131	1828908143280	832861757763	286997271876	73668882861	13660159464	1729840455	133923132	4782969															17565568854912
8	1406408618241	12274111577376	49933317553416	125682844994784	219049319021148	280290877596576	272367379973304	205018390649952	120809569170438	55914106540896	20258730741816	5685840492192	1211870421468	189635154912	20547634824	1377495072	43046721													1370114370683136
9	46411484401953	455676392310084	2102325173612433	6055600320947232	12202609805490852	18266520050401392	21050039003786100	19093450468856544	13818684265782462	8039475521834904	3768732072486126	1420174001815776	427010558303700	101057859714672	18411771319236	2491888585248	235939077801	13947137604	387420489											106868920913284608
10	1531578985264449	16708134384703080	86198784211990890	279619604372097000	639608661697810365	1096589517549441696	1462069497188809080	1552283950062178080	1332831148683731730	934616380008429360	538157435570140860	254895376365935280	99136201141765170	31488855485859360	8088766427996280	1654576828237728	263199920509485	31381059609000	2638333530090	139471376040	3486784401									8335775831236199424
11	50542106513726817	606505278164721804	3459836927712390291	12480976385373200760	31949180651331152055	61729942339601622108	93481993994290893981	113729177908793978784	113024275583500839690	92794817443452113880	63400865392271690910	36196762520676799440	17291145106983188430	6902093859430322520	2292753899890700730	629196326112445344	141049261045337733	25401963559578012	3585582437002335	382012098973560	28881035193483	1380766622796	31381059609							650190514836423555072
12	1667889514952984961	21834190013929984944	136463687587062405900	541884897618444171792	1534776643716774757794	3299587388123133810384	5594726351899238265756	7673984135841088082160	8665389444920728991823	8154665497819095795936	6450060466160749456920	4312077674295184279200	2444299027805255216220	1176021183898686621600	479756563598733430680	165421463892648825312	47940478453560484143	11578805130110240880	2302238257973925084	370305039034413648	46975751657454114	4523391456279696	310672490129100	13556617751088	282429536481					50714860157241037295616
13	55040353993448503713	780572292997996961748	5304343536509115717333	22991402084668274146032	71383658745201118844814	169015307060433304569384	317224105378558652763918	484347911644236688571952	612573817739756372079723	650123838667746197710956	584408458309078741658463	447892729513899923826144	293947208387646801298740	165612163138302504809520	80167420469358218536020	33314335253099994334176	11855017561491454564071	3596750968655654805996	924275153279152556739	199309889746189121328	35601325531700202846	5173144329035300904	595876049517416382	52342101136950768	3293410824904941	132177023073108	2541865828329			3955759092264800909058048
14	1816331681783800622529	27740338412698045871352	203639302438669745828334	956697786745363185298776	3231090305468276832592011	8354239466583553796517264	17195399460374844575310444	28924686641431821963499920	40506477753011508942096393	47865875319464984116479144	48203101675896268941041202	41670279121095247965366024	31084072264074450375525771	20078335451144366819904096	11253087623956532596520040	5475909668493918223610208	2312038432865041763468859	845302431457228921912008	266679272983238698464882	72221890597574626300392	16668476152921288072377	3246150489715074324240	526309035264647398764	69737126081464073232	7355882643808838139	594003541690547352	34482951827111214	1281100377477816	22876792454961	308549209196654470906527744

Technical details

Node id: n-67a731b2b62168b7 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-38e8cf17451719bf

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:             33,              36,                9;
  2:           1089,            2376,             1890,              648,                81;
  3:          35937,          117612,           157707,           110808,             43011,              8748,               729;
  4:        1185921,         5174928,          9761796,         10392624,           6827814,           2834352,            726084,            104976,              6561;
  5:       39135393,       213465780,        519109965,        740955600,         687308490,         432868536,         187447770,          55112400,          10530405,          1180980,            59049;
  6:     1291467969,      8453244888,      25167615462,      45060685560,       54027571455,       45696367728,       27954820116,       12462645744,        4018579695,        914078520,        139237542,         12754584,          531441;
  7:    42618442977,    325449928188,    1146471337935,    2469115984104,     3631603077333,     3858518877444,     3053826445131,     1828908143280,      832861757763,     286997271876,      73668882861,      13660159464,      1729840455,       133923132,        4782969;
  8:  1406408618241,  12274111577376,   49933317553416,  125682844994784,   219049319021148,   280290877596576,   272367379973304,   205018390649952,   120809569170438,   55914106540896,   20258730741816,    5685840492192,   1211870421468,    189635154912,    20547634824,    1377495072,     43046721;
  9: 46411484401953, 455676392310084, 2102325173612433, 6055600320947232, 12202609805490852, 18266520050401392, 21050039003786100, 19093450468856544, 13818684265782462, 8039475521834904, 3768732072486126, 1420174001815776, 427010558303700, 101057859714672, 18411771319236, 2491888585248, 235939077801, 13947137604, 387420489;

%F Formula

The row polynomials satisfy P(n,x) = (33+36*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[(33 + 36*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] = (33 + 36*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 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, 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_A013610 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)

Related ?

Same fingerprint ?

A272867 → basis A013609 → 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.