A200536 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,x^{2} + 3\,n x + 4\,n x^{2}\right)\,P_{n}(x) = \left(1 + 6\,x + 13\,x^{2} + 12\,x^{3} + 4\,x^{4} + 3\,n x + 13\,n x^{2} + 18\,n x^{3} + 8\,n x^{4}\right)\,P_{n-1}(x)\)

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

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	1	6	6																											13
2	1	12	39	48	20																									120
3	1	18	99	252	330	216	56																							972
4	1	24	186	720	1605	2160	1736	768	144																					7344
5	1	30	300	1560	4925	10098	13790	12480	7200	2400	352																			53136
6	1	36	441	2880	11775	32508	62923	86688	84780	57600	25872	6912	832																	373248
7	1	42	609	4788	24045	83538	209671	389112	539154	556920	423192	229824	84448	18816	1920															2566080
8	1	48	804	7392	44030	184464	569380	1330752	2391561	3326880	3578960	2951424	1831648	827904	257280	49152	4352													17356032
9	1	54	1026	10800	74430	365148	1338288	3774816	8346969	14625630	20403702	22648896	19883136	13632192	7145280	2764800	744192	124416	9728											115893504
10	1	60	1275	15120	118350	665496	2824710	9336960	24528285	51885900	89071983	124526160	141718980	130717440	96847200	56788992	25752960	8709120	2067200	307200	21504									765904896
11	1	66	1551	20460	179300	1136916	5485326	20796336	63304065	156989250	320253483	540560628	756962778	879139800	844054200	665482752	426288192	218287872	87211520	26188800	5558784	743424	47104							5018692608
12	1	72	1854	26928	261195	1843776	9966572	42634944	147596823	419993640	993061278	1965437136	3270747597	4586019984	5416697880	5375918592	4460703984	3069715968	1731335936	786677760	280836864	75829248	14555136	1769472	102400					32651735040
13	1	78	2184	34632	368355	2864862	17151134	81722784	317118087	1018570410	2739832524	6223440600	12005654037	19739070666	27705355470	33191683200	33874293024	29334827808	21423088544	13075645440	6586035456	2689150464	867549696	212779008	37273600	4153344	221184			211147886592
14	1	84	2541	43680	505505	4294836	28209545	148166304	637035399	2279937660	6876292423	17634734208	38707803771	73047929724	118857723795	166966696512	202471588956	211616810496	190035717872	145916010240	95130619584	52154539008	23728257280	8795824128	2588185600	581468160	93671424	9633792	475136	1358312177664

Technical details

Node id: n-23fe4c81851902da · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9d74e40d8b613922

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,  6,    6;
  2: 1, 12,   39,    48,    20;
  3: 1, 18,   99,   252,   330,    216,      56;
  4: 1, 24,  186,   720,  1605,   2160,    1736,     768,     144;
  5: 1, 30,  300,  1560,  4925,  10098,   13790,   12480,    7200,     2400,      352;
  6: 1, 36,  441,  2880, 11775,  32508,   62923,   86688,   84780,    57600,    25872,     6912,      832;
  7: 1, 42,  609,  4788, 24045,  83538,  209671,  389112,  539154,   556920,   423192,   229824,    84448,    18816,    1920;
  8: 1, 48,  804,  7392, 44030, 184464,  569380, 1330752, 2391561,  3326880,  3578960,  2951424,  1831648,   827904,  257280,   49152,   4352;
  9: 1, 54, 1026, 10800, 74430, 365148, 1338288, 3774816, 8346969, 14625630, 20403702, 22648896, 19883136, 13632192, 7145280, 2764800, 744192, 124416, 9728;

%F Formula

The row polynomials satisfy (1+3*n*x+(4*n-2)*x^2)*P(n,x) = (1+(3*n+6)*x+(13*n+13)*x^2+(18*n+12)*x^3+(8*n+4)*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[((1 + (3*n + 6)*x + (13*n + 13)*x^2 + (18*n + 12)*x^3 + (8*n + 4)*x^4)*P[n - 1, x])/(1 + 3*n*x + (4*n - 2)*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] = ((1 + (3*n + 6)*x + (13*n + 13)*x^2 + (18*n + 12)*x^3 + (8*n + 4)*x^4)*v[i - 1])/(1 + 3*n*x + (4*n - 2)*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 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.