A090452 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(8\,x - 8\,x^{2} + 8\,x^{3} - 2\,n - 12\,n x + 6\,n x^{2} - 28\,n x^{3} - 3\,n^{2} - 6\,n^{2} x - 5\,n^{2} x^{2} + 16\,n^{2} x^{3} - n^{3} + 5\,n^{3} x + 15\,n^{3} x^{2} + 16\,n^{3} x^{3} + 4\,n^{4} x + 10\,n^{4} x^{2} + n^{5} x\right)\,P_{n}(x) = \left(-24\,x - 48\,x^{2} + 48\,x^{4} + 24\,x^{5} - 2\,n - 28\,n x - 122\,n x^{2} - 12\,n x^{3} + 180\,n x^{4} + 96\,n x^{5} - 3\,n^{2} - 28\,n^{2} x + 21\,n^{2} x^{2} + 94\,n^{2} x^{3} + 144\,n^{2} x^{4} + 96\,n^{2} x^{5} - n^{3} + 3\,n^{3} x + 34\,n^{3} x^{2} + 126\,n^{3} x^{3} + 96\,n^{3} x^{4} + 4\,n^{4} x + 24\,n^{4} x^{2} + 20\,n^{4} x^{3} + n^{5} x + n^{5} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 8\,x^{2} + 8\,x^{3} + 32\,x^{4} + 28\,x^{5} + 8\,x^{6} - 3\,n x - 16\,n x^{2} + 18\,n x^{3} + 104\,n x^{4} + 105\,n x^{5} + 32\,n x^{6} - 2\,n^{2} x + 10\,n^{2} x^{2} + 46\,n^{2} x^{3} + 86\,n^{2} x^{4} + 84\,n^{2} x^{5} + 32\,n^{2} x^{6} + 6\,n^{3} x^{2} + 32\,n^{3} x^{3} + 46\,n^{3} x^{4} + 20\,n^{3} x^{5} + 2\,n^{4} x^{2} + 4\,n^{4} x^{3} + 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

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	1	12	18																											31
2	1	28	144	240	125																									538
3	1	48	459	1680	2850	2268	686																							7992
4	1	72	1062	6336	19275	32760	31556	16128	3402																					110592
5	1	100	2070	17760	80875	218988	371420	399360	264870	99000	15972																			1470416
6	1	132	3618	41600	259125	989388	2472736	4181184	4829625	3762000	1892682	555984	72501																	19060576
7	1	168	5859	86240	695625	3498768	11796799	27807552	46873890	56941500	49529172	30126096	12180168	2942940	321750															242806528
8	1	208	8964	163520	1645700	10435824	45005716	138322432	311826915	524106000	660910712	623278656	433410978	215815600	72844200	14936064	1405118													3054116608
9	1	252	13122	289536	3538800	27433728	145986288	560454912	1603306467	3489568500	5849735562	7592180544	7617310974	5855594472	3386026800	1425715200	412721478	73465704	6063356											38049405696
10	1	300	18540	485520	7059375	65384928	418309080	1946712960	6822116865	18430900500	38990943684	65209201200	86590613655	91243446840	75851802000	49142366208	24297683670	8856698760	2243441720	352716000	25924626									470495826432
11	1	352	25443	778800	13252250	144058068	1086329706	5993437824	25083632475	81671799000	210575458929	435156787728	726244390872	982472221500	1077325474500	953800962048	676047774006	378173816064	163215049250	52428068000	11801687436	1661183832	110038348							5783006226432
12	1	408	34074	1203840	23656875	297529848	2603669684	16738900992	82066726665	315078555000	965571439734	2394596691072	4852088881731	8082690459480	11104200657000	12584864421888	11734739106354	8948239538832	5523607709180	2717107008000	1040030756226	298538465808	60458212344	7702387200	464312500					70731710324736
13	1	468	44694	1803360	40475175	582079212	5836802972	43128214272	243974292705	1086032075700	3881535029658	11305659399840	27131094596151	54062200151820	89907652387800	125131270078464	145816214592906	142007548772232	115070521751900	77005783483200	42078553038522	18464058292680	6347735021784	1646599219200	303010337500	35254468080	1949666760			861576236077056
14	1	532	57582	2629536	66778075	1087346988	12363540644	103818000000	669378708999	3408612507300	13998188218014	47096741817216	131369723358237	306504938743500	602006489755200	999663886572288	1406811634519473	1678785878598192	1696584092265062	1447263456438800	1036279203170211	617575702790904	302617671256628	119819854365120	37378618834375	8842995743400	1491007654710	159615928080	8153364645	10458453182963712

Technical details

Node id: n-adf48e146cd7d0ad · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-7149774de98d320a

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,  12,    18;
  2: 1,  28,   144,    240,     125;
  3: 1,  48,   459,   1680,    2850,     2268,       686;
  4: 1,  72,  1062,   6336,   19275,    32760,     31556,     16128,       3402;
  5: 1, 100,  2070,  17760,   80875,   218988,    371420,    399360,     264870,      99000,      15972;
  6: 1, 132,  3618,  41600,  259125,   989388,   2472736,   4181184,    4829625,    3762000,    1892682,     555984,      72501;
  7: 1, 168,  5859,  86240,  695625,  3498768,  11796799,  27807552,   46873890,   56941500,   49529172,   30126096,   12180168,    2942940,     321750;
  8: 1, 208,  8964, 163520, 1645700, 10435824,  45005716, 138322432,  311826915,  524106000,  660910712,  623278656,  433410978,  215815600,   72844200,   14936064,   1405118;
  9: 1, 252, 13122, 289536, 3538800, 27433728, 145986288, 560454912, 1603306467, 3489568500, 5849735562, 7592180544, 7617310974, 5855594472, 3386026800, 1425715200, 412721478, 73465704, 6063356;

%F Formula

The row polynomials satisfy -(n^3+3*n^2+2*n-(n^5+4*n^4+5*n^3-6*n^2-12*n+8)*x-(10*n^4+15*n^3-5*n^2+6*n-8)*x^2-(16*n^3+16*n^2-28*n+8)*x^3)*P(n,x) = -(n^3+3*n^2+2*n-(n^5+4*n^4+3*n^3-28*n^2-28*n-24)*x-(n^5+24*n^4+34*n^3+21*n^2-122*n-48)*x^2-(20*n^4+126*n^3+94*n^2-12*n)*x^3-(96*n^3+144*n^2+180*n+48)*x^4-(96*n^2+96*n+24)*x^5)*P(n-1,x) - ((2*n^2+3*n+4)*x-(2*n^4+6*n^3+10*n^2-16*n-8)*x^2-(4*n^4+32*n^3+46*n^2+18*n+8)*x^3-(2*n^4+46*n^3+86*n^2+104*n+32)*x^4-(20*n^3+84*n^2+105*n+28)*x^5-(32*n^2+32*n+8)*x^6)*d/dx 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[(-(n^3 + 3*n^2 + 2*n - (n^5 + 4*n^4 + 3*n^3 - 28*n^2 - 28*n - 24)*x - (n^5 + 24*n^4 + 34*n^3 + 21*n^2 - 122*n - 48)*x^2 - (20*n^4 + 126*n^3 + 94*n^2 - 12*n)*x^3 - (96*n^3 + 144*n^2 + 180*n + 48)*x^4 - (96*n^2 + 96*n + 24)*x^5)*P[n - 1, x] - ((2*n^2 + 3*n + 4)*x - (2*n^4 + 6*n^3 + 10*n^2 - 16*n - 8)*x^2 - (4*n^4 + 32*n^3 + 46*n^2 + 18*n + 8)*x^3 - (2*n^4 + 46*n^3 + 86*n^2 + 104*n + 32)*x^4 - (20*n^3 + 84*n^2 + 105*n + 28)*x^5 - (32*n^2 + 32*n + 8)*x^6)*D[P[n - 1, x], x])/(-n^3 - 3*n^2 - 2*n + (n^5 + 4*n^4 + 5*n^3 - 6*n^2 - 12*n + 8)*x + (10*n^4 + 15*n^3 - 5*n^2 + 6*n - 8)*x^2 + (16*n^3 + 16*n^2 - 28*n + 8)*x^3)]];
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] = (-(n^3 + 3*n^2 + 2*n - (n^5 + 4*n^4 + 3*n^3 - 28*n^2 - 28*n - 24)*x - (n^5 + 24*n^4 + 34*n^3 + 21*n^2 - 122*n - 48)*x^2 - (20*n^4 + 126*n^3 + 94*n^2 - 12*n)*x^3 - (96*n^3 + 144*n^2 + 180*n + 48)*x^4 - (96*n^2 + 96*n + 24)*x^5)*v[i - 1] - ((2*n^2 + 3*n + 4)*x - (2*n^4 + 6*n^3 + 10*n^2 - 16*n - 8)*x^2 - (4*n^4 + 32*n^3 + 46*n^2 + 18*n + 8)*x^3 - (2*n^4 + 46*n^3 + 86*n^2 + 104*n + 32)*x^4 - (20*n^3 + 84*n^2 + 105*n + 28)*x^5 - (32*n^2 + 32*n + 8)*x^6)*deriv(v[i - 1], x))/(-n^3 - 3*n^2 - 2*n + (n^5 + 4*n^4 + 5*n^3 - 6*n^2 - 12*n + 8)*x + (10*n^4 + 15*n^3 - 5*n^2 + 6*n - 8)*x^2 + (16*n^3 + 16*n^2 - 28*n + 8)*x^3)); 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 Scaled array A078740 ((3,2)-Stirling2).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.