A094262 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6\,x^{2} - 12\,x^{3} - 6\,x^{4}\right)\,P_{n-2}(x) + \left(-2\,x^{2} - 6\,x^{3} - 6\,x^{4} - 2\,x^{5}\right)\,P_{n-2}'(x) + \left(1 + 8\,x + 7\,x^{2}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 8\,x + 9\,x^{2}\)

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	1	8	9																											18
2	1	24	108	160	75																									368
3	1	56	549	1984	3275	2520	735																							9120
4	1	120	2160	14240	45750	80136	78400	40320	8505																					269632
5	1	248	7569	80960	424750	1267776	2306038	2610944	1800225	693000	114345																			9305856
6	1	504	24948	406560	3198825	14451696	41011824	76578048	95757795	79571800	42231420	12972960	1756755																	367963136
7	1	1016	79389	1896384	21284025	136432296	554475621	1516706304	2887091019	3880894600	3677593535	2407204800	1037071035	264864600	30405375															16416000000
8	1	2040	247320	8440480	130786500	1142151192	6318306960	23727911040	63113196870	121925353000	173186513016	181057425120	137786068420	74297463240	26918892000	5881075200	585810225													816079642624
9	1	4088	760329	36416320	761499500	8812197072	64164992556	317540990208	1121620808070	2918367904000	5704167054806	8462751979968	9551777639404	8152215671600	5176675003500	2371510901760	741049934625	141421480200	12439852425											44745315090432
10	1	8184	2317788	153808160	4269037575	64166522112	600550195168	3807627411456	17267272952490	58077848263600	148449909521496	292922986284864	450177371802158	540378801842880	505004857156800	363710434447360	198005951948805	78815762265240	21645343219500	3666482820000	288735522075									2682888527347712
11	1	16376	7027029	640021184	23287104775	447857547192	5293741128907	42126951477248	239459135341338	1012091002001200	3270581824632386	8238085390613376	16383700375221374	25924291213051504	32735922923053350	32938078451998720	26234900875428245	16331341010264280	7777363945376025	2736173912472000	670155146736075	102038216880600	7273385294175							174649349292687360
12	1	32760	21228480	2635170720	124494347250	3028239782568	44599408133280	438592596773760	3069459132835095	15968811032174000	63688397361196608	199060913797462080	495275539897991580	991527052241209680	1607883677792496000	2118439606520380416	2266593513944895135	1960972919030525400	1359943985055335680	745110122961204000	315255238143095250	99351210502744200	21955925741349600	3035847774960000	197646339515625					12267816214664839168
13	1	65528	63980289	10767382080	655588274250	19985873832288	362877151317954	4356574032974592	37022711145726735	233380720733303000	1128284770737667543	4286159884435089024	13023949801609274892	32067169423915946880	64560844212435517500	106908243623482546176	146039279381414049423	164593126164957281064	152627518270935682975	115740177808431176000	71044732076720572170	34749156869698471200	13224305894776826850	3773761174126944000	759686646984890625	96198426369045000	5763367260275625			924903501333648113664
14	1	131064	192530628	43752508960	3413066816325	129452620581648	2873000763574032	41701242827821824	425624031515955285	3210991297094695400	18556695703041712788	84335147716548654432	307372741098084594569	911669828680404255840	2224557624752343230400	4500579811797506574336	7588600315541936791299	10694241912530417561928	12603123069269066084140	12398850403990697119200	10138646244304138255695	6840217594586234818960	3764928211423664965680	1662731609694835680000	574898365772089734375	149879286025780311000	27702585297724837500	3235454766163620000	179518217255251875	74497987743649768669184

Technical details

Node id: n-4b763390b67311c8 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-76f2941a93f2375f

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: 1,    8,      9;
  2: 1,   24,    108,      160,        75;
  3: 1,   56,    549,     1984,      3275,       2520,         735;
  4: 1,  120,   2160,    14240,     45750,      80136,       78400,        40320,          8505;
  5: 1,  248,   7569,    80960,    424750,    1267776,     2306038,      2610944,       1800225,        693000,        114345;
  6: 1,  504,  24948,   406560,   3198825,   14451696,    41011824,     76578048,      95757795,      79571800,      42231420,      12972960,       1756755;
  7: 1, 1016,  79389,  1896384,  21284025,  136432296,   554475621,   1516706304,    2887091019,    3880894600,    3677593535,    2407204800,    1037071035,     264864600,      30405375;
  8: 1, 2040, 247320,  8440480, 130786500, 1142151192,  6318306960,  23727911040,   63113196870,  121925353000,  173186513016,  181057425120,  137786068420,   74297463240,   26918892000,    5881075200,    585810225;
  9: 1, 4088, 760329, 36416320, 761499500, 8812197072, 64164992556, 317540990208, 1121620808070, 2918367904000, 5704167054806, 8462751979968, 9551777639404, 8152215671600, 5176675003500, 2371510901760, 741049934625, 141421480200, 12439852425;

%F Formula

The row polynomials satisfy P(n,x) = -(6*x^2+12*x^3+6*x^4)*P(n-2,x) - (2*x^2+6*x^3+6*x^4+2*x^5)*d/dx P(n-2,x) + (1+8*x+7*x^2)*P(n-1,x) + (x+2*x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x + 9*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 8*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[-(6*x^2 + 12*x^3 + 6*x^4)*P[n - 2, x] - (2*x^2 + 6*x^3 + 6*x^4 + 2*x^5)*D[P[n - 2, x], x] + (1 + 8*x + 7*x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 8*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(6*x^2 + 12*x^3 + 6*x^4)*v[i - 2] - (2*x^2 + 6*x^3 + 6*x^4 + 2*x^5)*deriv(v[i - 2], x) + (1 + 8*x + 7*x^2)*v[i - 1] + (x + 2*x^2 + x^3)*deriv(v[i - 1], x)); 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 Triangle read by rows: T(n,k) is the number of rooted trees with k nodes which are disjoint sets of labels with union {…?
  - 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.