A200536 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 6\,x^{2} + 7\,n x + 12\,n x^{2}\right)\,P_{n}(x) = \left(12 + 84\,x + 219\,x^{2} + 252\,x^{3} + 108\,x^{4} + 42\,n x + 219\,n x^{2} + 378\,n x^{3} + 216\,n x^{4}\right)\,P_{n-1}(x)\)

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

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	6	42	54																											102
2	36	504	1971	3024	1620																									7155
3	216	4536	29646	91476	148230	122472	40824																							437400
4	1296	36288	332424	1542240	4180005	6940080	6980904	3919104	944784																					24877125
5	7776	272160	3207600	19867680	75536550	188617086	317253510	357618240	259815600	110224800	20785248																			1353206250
6	46656	1959552	28238544	218816640	1070973900	3571650936	8428160727	14286603744	17349777180	14770123200	8386847568	2857026816	442158912																	71412384375
7	279936	13716864	233676576	2174122944	13016936520	54326497176	165062954574	373715102376	636671396214	814897457640	773206029288	528311875392	246061434528	69997156992	9183300480															3686871937500
8	1679616	94058496	1849257216	20081488896	142182643680	712687102848	2649403708560	7507637456256	16469076048201	28153640460960	37470138163920	38485103553792	29943664759008	17079306306048	6740542552320	1645647446016	187339329792													187208396015625
9	10077696	634894848	14149085184	175654241280	1435712662080	8401325438016	36940258891872	125736004211328	337497162734214	722122126656210	1237489596692118	1697436056852928	1852290124435296	1587850507785024	1046634530656320	512207767572480	175349612685312	37489905879552	3768826516992											9382839967968750
10	60466176	4232632320	105437894400	1472956047360	13649672361600	91330576227072	463653985667040	1842614049185280	5850399210252660	15036726651823800	31530896286063867	54132215946565680	76055189733284580	87063513824004480	80477084655064800	59182213395142656	33832077915461760	14496096940093440	4381260826003200	833109019545600	74979811759104									465358595185546875
11	362797056	27935373312	769220457984	11942372090880	123726267244800	931725014153472	5363494775077824	24375313780570368	89364430127526840	268188490923507000	665242545999071502	1371930760821342204	2358587208542162598	3379174985636188200	4021399355738707800	3948800832452399616	3165228130835212992	2037682605953643264	1028239516564634880	391769516441318400	105983963921493504	18145114445703168	1478173431822336							22882022631562500000
12	2176782336	182849716224	5514878048256	94198078808064	1078921484593920	9030696065943552	58126133431795968	297337157624973312	1236179962197222768	4242745541912834880	12152626879133031192	29264557931763624288	59514471311191486317	102425952761172685008	149145875334814473720	183286607410634466816	189135534216175083504	162569090931454158336	115015010594829421824	65833774320728494080	29730976213766978304	10197554318485180416	2496634926347925504	388695344159195136	28920784535654400					1117030881589892578125
13	13060694016	1188523155456	38966580596736	726187647983616	9111674817722880	83916345935245824	597197675618159616	3395786853695588352	15786874924352037792	60990758415974106720	198119142461323573776	545636285254219330080	1281375038061783124902	2575054092017941218342	4435528977906172355430	6547635423050631960960	8266971489977047305744	8892452577049024759776	8098666836192595387296	6188821540860209771520	3918213949730745240576	2018775534164208787968	824987439011301456384	257283926554707247104	57523440441416601600	8211189145362997248	562220051373121536			54196683514176269531250
14	78364164096	7679688081408	271943240454144	5490976978206720	74901472083809280	752719347513649152	5868750734529096960	36721457639455260672	188767574457547909824	810710169605542926720	2944912622512688139312	9129947538787951284864	24316010202990526350876	55887475864766579297208	111165064805277745677855	191614202964913986161856	286180735465965425514156	369762875320912027036992	412020047822457004218288	394005142428293862385920	321093644152288994610624	220852026608093801496576	126516013156092045507840	59263099668444625035264	22114285125384270873600	6322615641929507880960	1300696088851716673536	171414646774205054976	10869587659880349696	2615713981056331787109375

Technical details

Node id: n-c4e761abf0fd75f8 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-a42848b2e2dea42b

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:        6,        42,          54;
  2:       36,       504,        1971,         3024,          1620;
  3:      216,      4536,       29646,        91476,        148230,        122472,          40824;
  4:     1296,     36288,      332424,      1542240,       4180005,       6940080,        6980904,         3919104,          944784;
  5:     7776,    272160,     3207600,     19867680,      75536550,     188617086,      317253510,       357618240,       259815600,       110224800,         20785248;
  6:    46656,   1959552,    28238544,    218816640,    1070973900,    3571650936,     8428160727,     14286603744,     17349777180,     14770123200,       8386847568,       2857026816,        442158912;
  7:   279936,  13716864,   233676576,   2174122944,   13016936520,   54326497176,   165062954574,    373715102376,    636671396214,    814897457640,     773206029288,     528311875392,     246061434528,      69997156992,       9183300480;
  8:  1679616,  94058496,  1849257216,  20081488896,  142182643680,  712687102848,  2649403708560,   7507637456256,  16469076048201,  28153640460960,   37470138163920,   38485103553792,   29943664759008,   17079306306048,    6740542552320,   1645647446016,    187339329792;
  9: 10077696, 634894848, 14149085184, 175654241280, 1435712662080, 8401325438016, 36940258891872, 125736004211328, 337497162734214, 722122126656210, 1237489596692118, 1697436056852928, 1852290124435296, 1587850507785024, 1046634530656320, 512207767572480, 175349612685312, 37489905879552, 3768826516992;

%F Formula

The row polynomials satisfy (2+7*n*x+(12*n-6)*x^2)*P(n,x) = (12+(42*n+84)*x+(219*n+219)*x^2+(378*n+252)*x^3+(216*n+108)*x^4)*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[Together[((12 + (42*n + 84)*x + (219*n + 219)*x^2 + (378*n + 252)*x^3 + (216*n + 108)*x^4)*P[n - 1, x])/(2 + 7*n*x + (12*n - 6)*x^2)]];
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] = ((12 + (42*n + 84)*x + (219*n + 219)*x^2 + (378*n + 252)*x^3 + (216*n + 108)*x^4)*v[i - 1])/(2 + 7*n*x + (12*n - 6)*x^2)); 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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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

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.