A109983 → θ+1

nonnegative coefficients

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

Recurrence

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

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

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	0	2	6																											8
2	0	0	3	24	30																									57
3	0	0	0	4	60	180	140																							384
4	0	0	0	0	5	120	630	1120	630																					2505
5	0	0	0	0	0	6	210	1680	5040	6300	2772																			16008
6	0	0	0	0	0	0	7	336	3780	16800	34650	33264	12012																	100849
7	0	0	0	0	0	0	0	8	504	7560	46200	138600	216216	168168	51480															628736
8	0	0	0	0	0	0	0	0	9	720	13860	110880	450450	1009008	1261260	823680	218790													3888657
9	0	0	0	0	0	0	0	0	0	10	990	23760	240240	1261260	3783780	6726720	7001280	3938220	923780											23900040
10	0	0	0	0	0	0	0	0	0	0	11	1320	38610	480480	3153150	12108096	28588560	42007680	37413090	18475600	3879876									146146473
11	0	0	0	0	0	0	0	0	0	0	0	12	1716	60060	900900	7207200	34306272	102918816	199536480	249420600	193993800	85357272	16224936							889928064
12	0	0	0	0	0	0	0	0	0	0	0	0	13	2184	90090	1601600	15315300	88216128	325909584	798145920	1309458150	1422621200	981608628	389398464	67603900					5399971161
13	0	0	0	0	0	0	0	0	0	0	0	0	0	14	2730	131040	2722720	30630600	209513304	931170240	2793510720	5761615860	8180071900	7852869024	4867480800	1757701400	280816200			32668236552
14	0	0	0	0	0	0	0	0	0	0	0	0	0	0	15	3360	185640	4455360	58198140	465585120	2444321880	8779605120	22086194130	39264345120	49080431400	42184833600	23728968900	7862853600	1163381400	197123362785

Technical details

Node id: n-3f7b4a84cb4ddfc5 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-883c9e310e304e0b

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: 0, 2, 6;
  2: 0, 0, 3, 24, 30;
  3: 0, 0, 0,  4, 60, 180, 140;
  4: 0, 0, 0,  0,  5, 120, 630, 1120,  630;
  5: 0, 0, 0,  0,  0,   6, 210, 1680, 5040,  6300,  2772;
  6: 0, 0, 0,  0,  0,   0,   7,  336, 3780, 16800, 34650,  33264,  12012;
  7: 0, 0, 0,  0,  0,   0,   0,    8,  504,  7560, 46200, 138600, 216216,  168168,   51480;
  8: 0, 0, 0,  0,  0,   0,   0,    0,    9,   720, 13860, 110880, 450450, 1009008, 1261260,  823680,  218790;
  9: 0, 0, 0,  0,  0,   0,   0,    0,    0,    10,   990,  23760, 240240, 1261260, 3783780, 6726720, 7001280, 3938220, 923780;

%F Formula

The row polynomials satisfy n^2*P(n,x) = -((n^2-2*n-1)*x-(4*n+2)*x^2)*P(n-1,x) + ((2*n+1)*x^2+(2*n+1)*x^3)*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^2 - 2*n - 1)*x - (4*n + 2)*x^2)*P[n - 1, x] + ((2*n + 1)*x^2 + (2*n + 1)*x^3)*D[P[n - 1, x], x])/(n^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] = (-((n^2 - 2*n - 1)*x - (4*n + 2)*x^2)*v[i - 1] + ((2*n + 1)*x^2 + (2*n + 1)*x^3)*deriv(v[i - 1], x))/(n^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: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Related ?

Same fingerprint ?

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.