A281000 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n - 4\,n^{2} - n^{2} x - n^{3} x + 2\,n^{4} x\right)\,P_{n}(x) = \left(4 - 8\,x - 2\,x^{2} + 34\,n x + 7\,n x^{2} - 16\,n^{2} - 26\,n^{2} x - 5\,n^{2} x^{2} - 8\,n^{3} x - 6\,n^{3} x^{2} + 8\,n^{4} x + 6\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(8\,x + 2\,x^{2} - x^{3} - 32\,n x - 4\,n x^{2} + 3\,n x^{3} + 4\,n^{2} x^{2} - n^{2} x^{3} + 16\,n^{3} x^{2} - 4\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	2														5
3	10	20	3													33
4	35	140	63	4												242
5	126	840	756	144	5											1871
6	462	4620	6930	2640	275	6										14933
7	1716	24024	54054	34320	7150	468	7									121739
8	6435	120120	378378	360360	125125	16380	735	8								1007541
9	24310	583440	2450448	3267264	1701700	371280	33320	1088	9							8432859
10	92378	2771340	14965236	26604864	19399380	6348888	949620	62016	1539	10						71195271
11	352716	12932920	87297210	199536480	193993800	88884432	19942020	2170560	107730	2100	11					605219979
12	1352078	59491432	490804314	1402298040	1752872550	1070845776	336355404	54915168	4542615	177100	2783	12				5173657272
13	5200300	270415600	2677114440	9348653600	14607271250	11473347600	4805077200	1098303360	136278450	8855000	278300	3600	13			44430798713
14	20058300	1216870200	14237381340	59661407520	113936715750	111865139100	60235074900	18357356160	3188915730	310810500	16280550	421200	4563	14		383046435827
15	77558760	5429113200	74107395180	367008052320	841060119900	1009272143880	679317789150	266181664320	61652370780	8412604200	660990330	28501200	617526	5684	15	3313208926445

Technical details

Node id: n-ddf6aec16b6c760d · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-6aa82b38f20c7d91

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     1;
   2:     3,       2;
   3:    10,      20,        3;
   4:    35,     140,       63,        4;
   5:   126,     840,      756,      144,        5;
   6:   462,    4620,     6930,     2640,      275,       6;
   7:  1716,   24024,    54054,    34320,     7150,     468,      7;
   8:  6435,  120120,   378378,   360360,   125125,   16380,    735,     8;
   9: 24310,  583440,  2450448,  3267264,  1701700,  371280,  33320,  1088,    9;
  10: 92378, 2771340, 14965236, 26604864, 19399380, 6348888, 949620, 62016, 1539, 10;

%F Formula

The row polynomials satisfy -(4*n^2+2*n-(2*n^4-n^3-n^2)*x)*P(n,x) = -(16*n^2-4-(8*n^4-8*n^3-26*n^2+34*n-8)*x-(6*n^4-6*n^3-5*n^2+7*n-2)*x^2)*P(n-1,x) - ((32*n-8)*x-(16*n^3+4*n^2-4*n+2)*x^2+(4*n^3+n^2-3*n+1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n^2 - 4 - (8*n^4 - 8*n^3 - 26*n^2 + 34*n - 8)*x - (6*n^4 - 6*n^3 - 5*n^2 + 7*n - 2)*x^2)*P[n - 1, x] - ((32*n - 8)*x - (16*n^3 + 4*n^2 - 4*n + 2)*x^2 + (4*n^3 + n^2 - 3*n + 1)*x^3)*D[P[n - 1, x], x])/(-4*n^2 - 2*n + (2*n^4 - n^3 - n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(16*n^2 - 4 - (8*n^4 - 8*n^3 - 26*n^2 + 34*n - 8)*x - (6*n^4 - 6*n^3 - 5*n^2 + 7*n - 2)*x^2)*v[i - 1] - ((32*n - 8)*x - (16*n^3 + 4*n^2 - 4*n + 2)*x^2 + (4*n^3 + n^2 - 3*n + 1)*x^3)*deriv(v[i - 1], x))/(-4*n^2 - 2*n + (2*n^4 - n^3 - n^2)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by 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) = binomial(2*n+1, 2*k+1)*binomial(2*n-2*k, n-k)/(n+1-k) for 0 <= k <= n.?
  - 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
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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.