A110552 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 4\,x + 4\,x^{2} + 10\,n x - 18\,n x^{2} + 14\,n^{2} x^{2} - 8\,n^{2} x^{3} + n^{2} x^{4} + 6\,n^{3} x^{3} - 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(4 + 44\,x - 44\,x^{2} - 56\,x^{3} - 16\,x^{4} + 4\,x^{5} - 6\,n x + 118\,n x^{2} - 30\,n x^{3} - 32\,n x^{4} - 2\,n x^{5} - 26\,n^{2} x^{2} + 111\,n^{2} x^{3} + 9\,n^{2} x^{4} - 8\,n^{2} x^{5} - 27\,n^{3} x^{3} + 32\,n^{3} x^{4} + 4\,n^{3} x^{5} - 9\,n^{4} x^{4} + 3\,n^{4} x^{5} - n^{5} x^{5}\right)\,P_{n-1}(x) + \left(4\,x - 16\,x^{2} - 36\,x^{3} - 18\,x^{4} + 2\,x^{6} + 17\,n x^{2} - 6\,n x^{3} - 35\,n x^{4} - 12\,n x^{5} + 20\,n^{2} x^{3} + 13\,n^{2} x^{4} - 11\,n^{2} x^{5} - 4\,n^{2} x^{6} + 8\,n^{3} x^{4} + 8\,n^{3} x^{5} + n^{4} x^{5} + n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	8														9
2	1	20	36													57
3	1	40	153	128												322
4	1	76	459	784	400											1720
5	1	144	1206	3136	3225	1152										8864
6	1	276	2970	10400	16125	11556	3136									44464
7	1	536	7047	31040	64375	69336	37681	8192								218208
8	1	1052	16317	86576	224875	323316	263767	114752	20736							1051392
9	1	2080	37116	230272	718550	1292256	1406300	918016	331857	51200						4987648
10	1	4132	83268	591168	2153550	4649400	6326292	5507328	2986713	921700	123904					23347456
11	1	8232	184725	1476480	6149250	15491952	25298994	27532800	19910205	9217000	2478201	294912				108042752
12	1	16428	405999	3607120	16904250	48677112	92747886	121131648	109499445	67589500	27260211	6488208	692224			495020032
13	1	32816	885330	8654272	45068375	146009952	317960412	484492800	525573279	405526000	218079026	77858496	16613545	1605632		2248359936
14	1	65588	1917630	20451808	117163475	421770492	1033308276	1799466240	2277414711	2108692300	1417496366	674769888	215976085	41746628	3686400	10133925888

Technical details

Node id: n-afbb26c15fdf56cb · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-7edc59f261520df0

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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;
  2: 1,   20,    36;
  3: 1,   40,   153,    128;
  4: 1,   76,   459,    784,    400;
  5: 1,  144,  1206,   3136,   3225,    1152;
  6: 1,  276,  2970,  10400,  16125,   11556,    3136;
  7: 1,  536,  7047,  31040,  64375,   69336,   37681,   8192;
  8: 1, 1052, 16317,  86576, 224875,  323316,  263767, 114752,  20736;
  9: 1, 2080, 37116, 230272, 718550, 1292256, 1406300, 918016, 331857, 51200;

%F Formula

The row polynomials satisfy (4+(10*n-4)*x+(14*n^2-18*n+4)*x^2+(6*n^3-8*n^2)*x^3+(n^4-2*n^3+n^2)*x^4)*P(n,x) = (4-(6*n-44)*x-(26*n^2-118*n+44)*x^2-(27*n^3-111*n^2+30*n+56)*x^3-(9*n^4-32*n^3-9*n^2+32*n+16)*x^4-(n^5-3*n^4-4*n^3+8*n^2+2*n-4)*x^5)*P(n-1,x) + (4*x+(17*n-16)*x^2+(20*n^2-6*n-36)*x^3+(8*n^3+13*n^2-35*n-18)*x^4+(n^4+8*n^3-11*n^2-12*n)*x^5+(n^4-4*n^2+2)*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[((4 - (6*n - 44)*x - (26*n^2 - 118*n + 44)*x^2 - (27*n^3 - 111*n^2 + 30*n + 56)*x^3 - (9*n^4 - 32*n^3 - 9*n^2 + 32*n + 16)*x^4 - (n^5 - 3*n^4 - 4*n^3 + 8*n^2 + 2*n - 4)*x^5)*P[n - 1, x] + (4*x + (17*n - 16)*x^2 + (20*n^2 - 6*n - 36)*x^3 + (8*n^3 + 13*n^2 - 35*n - 18)*x^4 + (n^4 + 8*n^3 - 11*n^2 - 12*n)*x^5 + (n^4 - 4*n^2 + 2)*x^6)*D[P[n - 1, x], x])/(4 + (10*n - 4)*x + (14*n^2 - 18*n + 4)*x^2 + (6*n^3 - 8*n^2)*x^3 + (n^4 - 2*n^3 + n^2)*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] = ((4 - (6*n - 44)*x - (26*n^2 - 118*n + 44)*x^2 - (27*n^3 - 111*n^2 + 30*n + 56)*x^3 - (9*n^4 - 32*n^3 - 9*n^2 + 32*n + 16)*x^4 - (n^5 - 3*n^4 - 4*n^3 + 8*n^2 + 2*n - 4)*x^5)*v[i - 1] + (4*x + (17*n - 16)*x^2 + (20*n^2 - 6*n - 36)*x^3 + (8*n^3 + 13*n^2 - 35*n - 18)*x^4 + (n^4 + 8*n^3 - 11*n^2 - 12*n)*x^5 + (n^4 - 4*n^2 + 2)*x^6)*deriv(v[i - 1], x))/(4 + (10*n - 4)*x + (14*n^2 - 18*n + 4)*x^2 + (6*n^3 - 8*n^2)*x^3 + (n^4 - 2*n^3 + n^2)*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 A triangular array related to A077028 and distributing the values of A007582.?
  - 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.