A122076 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-200 + 130\,x - 82\,x^{2} - 4\,x^{3} + 200\,n - 630\,n x + 562\,n x^{2} + 28\,n x^{3} + 500\,n^{2} x - 1080\,n^{2} x^{2} + 160\,n^{2} x^{3} - 8\,n^{2} x^{4} + 600\,n^{3} x^{2} - 424\,n^{3} x^{3} + 40\,n^{3} x^{4} + 240\,n^{4} x^{3} - 64\,n^{4} x^{4} + 32\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-800 - 830\,x + 782\,x^{2} + 1160\,x^{3} + 300\,x^{4} - 24\,x^{5} + 800\,n - 1870\,n x - 3662\,n x^{2} - 1244\,n x^{3} + 176\,n x^{4} + 88\,n x^{5} + 2700\,n^{2} x - 120\,n^{2} x^{2} - 3140\,n^{2} x^{3} - 1100\,n^{2} x^{4} - 24\,n^{2} x^{5} + 3000\,n^{3} x^{2} + 2104\,n^{3} x^{3} - 448\,n^{3} x^{4} - 200\,n^{3} x^{5} + 1120\,n^{4} x^{3} + 944\,n^{4} x^{4} + 64\,n^{4} x^{5} + 128\,n^{5} x^{4} + 96\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-125 - 75\,x + 565\,x^{2} + 961\,x^{3} + 530\,x^{4} + 68\,x^{5} - 16\,x^{6} - 750\,n x - 1550\,n x^{2} - 760\,n x^{3} + 260\,n x^{4} + 268\,n x^{5} + 48\,n x^{6} - 1200\,n^{2} x^{2} - 2600\,n^{2} x^{3} - 1732\,n^{2} x^{4} - 316\,n^{2} x^{5} + 16\,n^{2} x^{6} - 600\,n^{3} x^{3} - 1240\,n^{3} x^{4} - 768\,n^{3} x^{5} - 128\,n^{3} x^{6} - 80\,n^{4} x^{4} - 144\,n^{4} x^{5} - 64\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 3 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	3	8														11
2	7	32	18													57
3	18	120	135	32												305
4	47	416	720	384	50											1617
5	123	1360	3195	2720	875	72										8345
6	322	4272	12690	14912	7875	1728	98									41897
7	843	13048	46872	70336	51975	19152	3087	128								205441
8	2207	39040	164520	300544	286000	149184	41160	5120	162							987937
9	5778	114984	556065	1197408	1395225	944136	372498	80640	8019	200						4674953
10	15127	334480	1825560	4527680	6245375	5201568	2679810	837120	147015	12000	242					21825977
11	39603	963248	5856345	16439456	26203375	25934832	16521967	6779520	1732104	253000	17303	288				100741041
12	103682	2751072	18436788	57784448	104518050	119894688	91085708	46413312	15661107	3352800	415272	24192	338			460441457
13	271443	7802600	57142800	197791360	400286250	522306720	461156150	281132800	118262430	33605000	6142565	655200	32955	392		2086588665
14	710647	21998816	174790350	662206720	1482649000	2168822880	2182982340	1550553600	782689635	278278000	67827760	10745280	999635	43904	450	9385299017

Technical details

Node id: n-d3d7c00ff700f502 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-ab269ce9163b66d2

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:    2;
  1:    3,      8;
  2:    7,     32,     18;
  3:   18,    120,    135,      32;
  4:   47,    416,    720,     384,      50;
  5:  123,   1360,   3195,    2720,     875,     72;
  6:  322,   4272,  12690,   14912,    7875,   1728,     98;
  7:  843,  13048,  46872,   70336,   51975,  19152,   3087,   128;
  8: 2207,  39040, 164520,  300544,  286000, 149184,  41160,  5120,  162;
  9: 5778, 114984, 556065, 1197408, 1395225, 944136, 372498, 80640, 8019, 200;

%F Formula

The row polynomials satisfy (200*n-200+(500*n^2-630*n+130)*x+(600*n^3-1080*n^2+562*n-82)*x^2+(240*n^4-424*n^3+160*n^2+28*n-4)*x^3+(32*n^5-64*n^4+40*n^3-8*n^2)*x^4)*P(n,x) = (800*n-800+(2700*n^2-1870*n-830)*x+(3000*n^3-120*n^2-3662*n+782)*x^2+(1120*n^4+2104*n^3-3140*n^2-1244*n+1160)*x^3+(128*n^5+944*n^4-448*n^3-1100*n^2+176*n+300)*x^4+(96*n^5+64*n^4-200*n^3-24*n^2+88*n-24)*x^5)*P(n-1,x) - (125+(750*n+75)*x+(1200*n^2+1550*n-565)*x^2+(600*n^3+2600*n^2+760*n-961)*x^3+(80*n^4+1240*n^3+1732*n^2-260*n-530)*x^4+(144*n^4+768*n^3+316*n^2-268*n-68)*x^5+(64*n^4+128*n^3-16*n^2-48*n+16)*x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 8*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((800*n - 800 + (2700*n^2 - 1870*n - 830)*x + (3000*n^3 - 120*n^2 - 3662*n + 782)*x^2 + (1120*n^4 + 2104*n^3 - 3140*n^2 - 1244*n + 1160)*x^3 + (128*n^5 + 944*n^4 - 448*n^3 - 1100*n^2 + 176*n + 300)*x^4 + (96*n^5 + 64*n^4 - 200*n^3 - 24*n^2 + 88*n - 24)*x^5)*P[n - 1, x] - (125 + (750*n + 75)*x + (1200*n^2 + 1550*n - 565)*x^2 + (600*n^3 + 2600*n^2 + 760*n - 961)*x^3 + (80*n^4 + 1240*n^3 + 1732*n^2 - 260*n - 530)*x^4 + (144*n^4 + 768*n^3 + 316*n^2 - 268*n - 68)*x^5 + (64*n^4 + 128*n^3 - 16*n^2 - 48*n + 16)*x^6)*D[P[n - 1, x], x])/(200*n - 200 + (500*n^2 - 630*n + 130)*x + (600*n^3 - 1080*n^2 + 562*n - 82)*x^2 + (240*n^4 - 424*n^3 + 160*n^2 + 28*n - 4)*x^3 + (32*n^5 - 64*n^4 + 40*n^3 - 8*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] = 2; v[2] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((800*n - 800 + (2700*n^2 - 1870*n - 830)*x + (3000*n^3 - 120*n^2 - 3662*n + 782)*x^2 + (1120*n^4 + 2104*n^3 - 3140*n^2 - 1244*n + 1160)*x^3 + (128*n^5 + 944*n^4 - 448*n^3 - 1100*n^2 + 176*n + 300)*x^4 + (96*n^5 + 64*n^4 - 200*n^3 - 24*n^2 + 88*n - 24)*x^5)*v[i - 1] - (125 + (750*n + 75)*x + (1200*n^2 + 1550*n - 565)*x^2 + (600*n^3 + 2600*n^2 + 760*n - 961)*x^3 + (80*n^4 + 1240*n^3 + 1732*n^2 - 260*n - 530)*x^4 + (144*n^4 + 768*n^3 + 316*n^2 - 268*n - 68)*x^5 + (64*n^4 + 128*n^3 - 16*n^2 - 48*n + 16)*x^6)*deriv(v[i - 1], x))/(200*n - 200 + (500*n^2 - 630*n + 130)*x + (600*n^3 - 1080*n^2 + 562*n - 82)*x^2 + (240*n^4 - 424*n^3 + 160*n^2 + 28*n - 4)*x^3 + (32*n^5 - 64*n^4 + 40*n^3 - 8*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 Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.?
  - 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.