A142458 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 3\,n\right)\,P_{n}(x) = \left(2 - 20\,x - 10\,x^{2} + 6\,n - 48\,n x - 9\,n x^{2} + 36\,n^{2} x + 54\,n^{2} x^{2} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(6\,x - 6\,x^{3} + 18\,n x - 9\,n x^{2} - 9\,n x^{3} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-4 + 8\,x - 3\,n - 24\,n x + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 6\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 2\,x\)

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	2														3
2	1	16	3													20
3	1	78	117	4												200
4	1	332	1638	664	5											2640
5	1	1354	16446	21928	3385	6										43120
6	1	5448	141525	438560	235875	16344	7									837760
7	1	21830	1121427	6838700	8548375	2242854	76405	8								18849600
8	1	87364	8472144	92310776	222257750	138466164	19768336	349456	9							481712000
9	1	349506	62188020	1137735408	4735566270	5682679524	1991036964	165834720	1572777	10						13776963200
10	1	1398080	448422615	13201536000	88583577450	181845744768	124017008430	26403072000	1345267845	6990400	11					435853017600
11	1	5592382	3197677665	146962780500	1514628746850	4929237482148	5750777062506	2423405994960	330666256125	10658925550	30758101	12				15109571276800
12	1	22369596	22643789418	1589007771160	24282760599675	118853998631208	218529528375228	158471998174944	43708969079415	3972519427900	83027227866	134217576	13			569514609664000
13	1	89478458	159647375322	16826021007544	371291159786375	2630146796089578	7202647003997004	8231596575996576	3945220194134367	742582319572750	46271557770746	638589501288	581609977	14		23187380536320000
14	1	357913912	1122497681673	175497557756704	5478793391265205	54557731706255448	213562907961889101	362190249343849344	274580881665285987	90929552843759080	12053345460783451	526492673270112	4864156620583	2505397384	15	1014061442121728000

Technical details

Node id: n-f9391d7b7df1563b · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-506bdfb5cd7829bb

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,      2;
  2: 1,     16,        3;
  3: 1,     78,      117,          4;
  4: 1,    332,     1638,        664,          5;
  5: 1,   1354,    16446,      21928,       3385,          6;
  6: 1,   5448,   141525,     438560,     235875,      16344,          7;
  7: 1,  21830,  1121427,    6838700,    8548375,    2242854,      76405,         8;
  8: 1,  87364,  8472144,   92310776,  222257750,  138466164,   19768336,    349456,       9;
  9: 1, 349506, 62188020, 1137735408, 4735566270, 5682679524, 1991036964, 165834720, 1572777, 10;

%F Formula

The row polynomials satisfy (3*n-2)*P(n,x) = (6*n+2+(36*n^2-48*n-20)*x-(27*n^3-54*n^2+9*n+10)*x^2)*P(n-2,x) + ((18*n+6)*x-(27*n^2+9*n)*x^2+(27*n^2-9*n-6)*x^3)*d/dx P(n-2,x) - (3*n+4-(18*n^2-24*n+8)*x)*P(n-1,x) + ((9*n-6)*x-(9*n-6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 2 + (36*n^2 - 48*n - 20)*x - (27*n^3 - 54*n^2 + 9*n + 10)*x^2)*P[n - 2, x] + ((18*n + 6)*x - (27*n^2 + 9*n)*x^2 + (27*n^2 - 9*n - 6)*x^3)*D[P[n - 2, x], x] - (3*n + 4 - (18*n^2 - 24*n + 8)*x)*P[n - 1, x] + ((9*n - 6)*x - (9*n - 6)*x^2)*D[P[n - 1, x], x])/(3*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; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 2 + (36*n^2 - 48*n - 20)*x - (27*n^3 - 54*n^2 + 9*n + 10)*x^2)*v[i - 2] + ((18*n + 6)*x - (27*n^2 + 9*n)*x^2 + (27*n^2 - 9*n - 6)*x^3)*deriv(v[i - 2], x) - (3*n + 4 - (18*n^2 - 24*n + 8)*x)*v[i - 1] + ((9*n - 6)*x - (9*n - 6)*x^2)*deriv(v[i - 1], x))/(3*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
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 T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257616 → θ+1

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.