A200536 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 3\,x - 12\,x^{2} - 6\,x^{3} + 4\,x^{4} + 9\,n x + 16\,n x^{2} - 6\,n x^{3} - 16\,n x^{4} + 9\,n^{2} x^{2} + 24\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n}(x) = \left(1 + 9\,x + 33\,x^{2} + 63\,x^{3} + 66\,x^{4} + 36\,x^{5} + 8\,x^{6} + 9\,n x + 61\,n x^{2} + 162\,n x^{3} + 210\,n x^{4} + 132\,n x^{5} + 32\,n x^{6} + 9\,n^{2} x^{2} + 51\,n^{2} x^{3} + 106\,n^{2} x^{4} + 96\,n^{2} x^{5} + 32\,n^{2} x^{6}\right)\,P_{n-1}(x)\)

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

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	24	117	192	100																									434
3	1	36	297	1008	1650	1296	392																							4680
4	1	48	558	2880	8025	12960	12152	6144	1296																					44064
5	1	60	900	6240	24625	60588	96530	99840	64800	24000	3872																			381456
6	1	72	1323	11520	58875	195048	440461	693504	763020	576000	284592	82944	10816																	3118176
7	1	84	1827	19152	120225	501228	1467697	3112896	4852386	5569200	4655112	2757888	1097824	263424	28800															24447744
8	1	96	2412	29568	220150	1106784	3985660	10646016	21524049	33268800	39368560	35417088	23811424	11590656	3859200	786432	73984													185690880
9	1	108	3078	43200	372150	2190888	9368016	30198528	75122721	146256300	224440722	271786752	258480768	190850688	107179200	44236800	12651264	2239488	184832											1375605504
10	1	120	3825	60480	591750	3992976	19772970	74695680	220754565	518859000	979791813	1494313920	1842346740	1830044160	1452708000	908623872	437800320	156764160	39276800	6144000	451584									9986996736
11	1	132	4653	81840	896500	6821496	38397282	166370688	569736585	1569892500	3522788313	6486727536	9840516114	12307957200	12660813000	10647724032	7246899264	3929181696	1657018880	523776000	116734464	16355328	1083392							71309776896
12	1	144	5562	107712	1305975	11062656	69766004	341079552	1328371407	4199936400	10923674058	23585245632	42519718761	64204279776	81250468200	86014697472	75831967728	55254887424	32895382784	15733555200	5897574144	1668243456	334768128	42467328	2560000					502111125504
13	1	156	6552	138528	1841775	17189172	120057938	653782272	2854062783	10185704100	30138157764	74681287200	156073502481	276346989324	415580332050	531066931200	575862981408	528026900544	407038682336	261512908800	138306744576	59161310208	19953643008	5106696192	931840000	107986944	5971968			3493735649280
14	1	168	7623	174720	2527525	25769016	197466815	1185330432	5733318591	22799376600	75639216653	211616810496	503201449023	1022671016136	1782865856925	2671467144192	3442017012252	3809102588928	3610678639568	2918320204800	1997743011264	1147399858176	545749917440	211099779072	64704640000	15118172160	2529128448	269746176	13778944	24062151942144

Technical details

Node id: n-43519ccf905be15d · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-6a29f8d72993234b

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,  24,  117,   192,    100;
  3: 1,  36,  297,  1008,   1650,    1296,     392;
  4: 1,  48,  558,  2880,   8025,   12960,   12152,     6144,     1296;
  5: 1,  60,  900,  6240,  24625,   60588,   96530,    99840,    64800,     24000,      3872;
  6: 1,  72, 1323, 11520,  58875,  195048,  440461,   693504,   763020,    576000,    284592,     82944,     10816;
  7: 1,  84, 1827, 19152, 120225,  501228, 1467697,  3112896,  4852386,   5569200,   4655112,   2757888,   1097824,    263424,     28800;
  8: 1,  96, 2412, 29568, 220150, 1106784, 3985660, 10646016, 21524049,  33268800,  39368560,  35417088,  23811424,  11590656,   3859200,   786432,    73984;
  9: 1, 108, 3078, 43200, 372150, 2190888, 9368016, 30198528, 75122721, 146256300, 224440722, 271786752, 258480768, 190850688, 107179200, 44236800, 12651264, 2239488, 184832;

%F Formula

The row polynomials satisfy (1+(9*n-3)*x+(9*n^2+16*n-12)*x^2+(24*n^2-6*n-6)*x^3+(16*n^2-16*n+4)*x^4)*P(n,x) = (1+(9*n+9)*x+(9*n^2+61*n+33)*x^2+(51*n^2+162*n+63)*x^3+(106*n^2+210*n+66)*x^4+(96*n^2+132*n+36)*x^5+(32*n^2+32*n+8)*x^6)*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[((1 + (9*n + 9)*x + (9*n^2 + 61*n + 33)*x^2 + (51*n^2 + 162*n + 63)*x^3 + (106*n^2 + 210*n + 66)*x^4 + (96*n^2 + 132*n + 36)*x^5 + (32*n^2 + 32*n + 8)*x^6)*P[n - 1, x])/(1 + (9*n - 3)*x + (9*n^2 + 16*n - 12)*x^2 + (24*n^2 - 6*n - 6)*x^3 + (16*n^2 - 16*n + 4)*x^4)]];
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] = ((1 + (9*n + 9)*x + (9*n^2 + 61*n + 33)*x^2 + (51*n^2 + 162*n + 63)*x^3 + (106*n^2 + 210*n + 66)*x^4 + (96*n^2 + 132*n + 36)*x^5 + (32*n^2 + 32*n + 8)*x^6)*v[i - 1])/(1 + (9*n - 3)*x + (9*n^2 + 16*n - 12)*x^2 + (24*n^2 - 6*n - 6)*x^3 + (16*n^2 - 16*n + 4)*x^4)); 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 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.