A269951 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-20 + 9\,n - n^{2}\right)\,P_{n-4}(x) + \left(20 - 16\,x - 13\,n + 4\,n x + 2\,n^{2}\right)\,P_{n-3}(x) + \left(16\,x^{2} - 4\,n x^{2}\right)\,P_{n-3}'(x) + \left(3 + 8\,x + 9\,x^{2} + 2\,n - 4\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-8\,x^{2} + 4\,x^{3} + 4\,n x^{2}\right)\,P_{n-2}'(x) + \left(-3 + 4\,x + 2\,n\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 4\,x\)

\(P_{3}(x) = 8\,x + 9\,x^{2}\)

\(P_{4}(x) = 20\,x + 45\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	0	4														4
3	0	8	9													17
4	0	20	45	16												81
5	0	64	207	144	25											440
6	0	260	1044	1040	350	36										2730
7	0	1304	6021	7520	3625	720	49									19239
8	0	7828	39861	58144	35375	10080	1323	64								152675
9	0	54800	300375	490896	352100	127260	24010	2240	81							1351762
10	0	438404	2547648	4545056	3688300	1581804	380044	51072	3564	100						13235992
11	0	3945640	24059889	46052544	41167850	20111112	5761371	989184	99630	5400	121					142192741
12	0	39456404	250607637	508445616	491608750	266267700	87328143	17946432	2315790	181500	7865	144				1664165981
13	0	434020448	2855469663	6086346640	6282072775	3702017880	1352743931	319526592	49506633	4983000	312785	11088	169			21087011604
14	0	5208245380	35340967620	78605995056	85769166450	54243472536	21663919200	5723284160	1022513382	123901800	10009120	514800	15210	196		287712004910
15	0	67707189944	472226462829	1090276157984	1248205324275	838492000632	360893334802	104587390336	20964639267	2937191400	286295438	18985824	815087	20384	225	4206595808427

Technical details

Node id: n-9fc9e57f912382e5 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-a9d50a5a25540ac9

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: 0,      4;
   3: 0,      8,       9;
   4: 0,     20,      45,      16;
   5: 0,     64,     207,     144,      25;
   6: 0,    260,    1044,    1040,     350,      36;
   7: 0,   1304,    6021,    7520,    3625,     720,     49;
   8: 0,   7828,   39861,   58144,   35375,   10080,   1323,    64;
   9: 0,  54800,  300375,  490896,  352100,  127260,  24010,  2240,   81;
  10: 0, 438404, 2547648, 4545056, 3688300, 1581804, 380044, 51072, 3564, 100;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-9*n+20)*P(n-4,x) + (2*n^2-13*n+20+(4*n-16)*x)*P(n-3,x) - (4*n-16)*x^2*d/dx P(n-3,x) - (n^2-2*n-3+(4*n-8)*x-9*x^2)*P(n-2,x) + ((4*n-8)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n-3+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 4*x, P(3,x) = 8*x + 9*x^2, P(4,x) = 20*x + 45*x^2 + 16*x^3.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 4*x;
P[3, x_] := 8*x + 9*x^2;
P[4, x_] := 20*x + 45*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 9*n + 20)*P[n - 4, x] + (2*n^2 - 13*n + 20 + (4*n - 16)*x)*P[n - 3, x] - (4*n - 16)*x^2*D[P[n - 3, x], x] - (n^2 - 2*n - 3 + (4*n - 8)*x - 9*x^2)*P[n - 2, x] + ((4*n - 8)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n - 3 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4*x; v[3] = 8*x + 9*x^2; v[4] = 20*x + 45*x^2 + 16*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 9*n + 20)*v[i - 4] + (2*n^2 - 13*n + 20 + (4*n - 16)*x)*v[i - 3] - (4*n - 16)*x^2*deriv(v[i - 3], x) - (n^2 - 2*n - 3 + (4*n - 8)*x - 9*x^2)*v[i - 2] + ((4*n - 8)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n - 3 + 4*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], 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

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) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>…?
  - 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.