chebyshev_t → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-9 + 6\,x + 9\,n - 28\,n x + 33\,n^{2} x - 4\,n^{2} x^{2} - 11\,n^{3} x + 8\,n^{3} x^{2} - 5\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(-9 - 30\,x - 20\,x^{2} - 4\,x^{3} + 9\,n + 8\,n x + n x^{2} + 33\,n^{2} x + 34\,n^{2} x^{2} + 7\,n^{2} x^{3} - 11\,n^{3} x - 11\,n^{3} x^{2} - n^{3} x^{3} - 5\,n^{4} x^{2} - 3\,n^{4} x^{3} + n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(18\,x + 35\,x^{2} + 21\,x^{3} + 4\,x^{4} + 22\,n x^{2} + 19\,n x^{3} + 4\,n x^{4} - 22\,n^{2} x^{2} - 17\,n^{2} x^{3} - 3\,n^{2} x^{4} - 4\,n^{3} x^{3} - 2\,n^{3} x^{4} + 2\,n^{4} x^{3} + n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 4\,x\)

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	4														5
2	1	16	18													35
3	1	36	108	64												209
4	1	64	360	512	200											1137
5	1	100	900	2240	2000	576										5817
6	1	144	1890	7168	10800	6912	1568									28483
7	1	196	3528	18816	42000	44352	21952	4096								134941
8	1	256	6048	43008	132000	202752	163072	65536	10368							623041
9	1	324	9720	88704	356400	741312	856128	552960	186624	25600						2817773
10	1	400	14850	168960	858000	2306304	3567200	3276800	1762560	512000	61952					12529027
11	1	484	21780	302016	1887600	6342336	12556544	15319040	11632896	5350400	1362944	147456				54923497
12	1	576	30888	512512	3861000	15814656	38811136	60162048	60279552	38912000	15611904	3538944	346112			237881329
13	1	676	42588	832832	7436000	36406656	108116736	206389248	261211392	221312000	124027904	44089344	8998912	802816		1019667105
14	1	784	57330	1304576	13613600	78414336	276529344	635043840	984566016	1048678400	768018944	379846656	121139200	22478848	1843200	4331535075

Technical details

Node id: n-c443535979497fee · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-ef846d88422ec39f

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,   4;
  2: 1,  16,   18;
  3: 1,  36,  108,    64;
  4: 1,  64,  360,   512,    200;
  5: 1, 100,  900,  2240,   2000,    576;
  6: 1, 144, 1890,  7168,  10800,   6912,   1568;
  7: 1, 196, 3528, 18816,  42000,  44352,  21952,   4096;
  8: 1, 256, 6048, 43008, 132000, 202752, 163072,  65536,  10368;
  9: 1, 324, 9720, 88704, 356400, 741312, 856128, 552960, 186624, 25600;

%F Formula

The row polynomials satisfy (9*n-9-(11*n^3-33*n^2+28*n-6)*x+(n^5-5*n^4+8*n^3-4*n^2)*x^2)*P(n,x) = (9*n-9-(11*n^3-33*n^2-8*n+30)*x+(n^5-5*n^4-11*n^3+34*n^2+n-20)*x^2+(n^5-3*n^4-n^3+7*n^2-4)*x^3)*P(n-1,x) + (18*x-(22*n^2-22*n-35)*x^2+(2*n^4-4*n^3-17*n^2+19*n+21)*x^3+(n^4-2*n^3-3*n^2+4*n+4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n - 9 - (11*n^3 - 33*n^2 - 8*n + 30)*x + (n^5 - 5*n^4 - 11*n^3 + 34*n^2 + n - 20)*x^2 + (n^5 - 3*n^4 - n^3 + 7*n^2 - 4)*x^3)*P[n - 1, x] + (18*x - (22*n^2 - 22*n - 35)*x^2 + (2*n^4 - 4*n^3 - 17*n^2 + 19*n + 21)*x^3 + (n^4 - 2*n^3 - 3*n^2 + 4*n + 4)*x^4)*D[P[n - 1, x], x])/(9*n - 9 - (11*n^3 - 33*n^2 + 28*n - 6)*x + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((9*n - 9 - (11*n^3 - 33*n^2 - 8*n + 30)*x + (n^5 - 5*n^4 - 11*n^3 + 34*n^2 + n - 20)*x^2 + (n^5 - 3*n^4 - n^3 + 7*n^2 - 4)*x^3)*v[i - 1] + (18*x - (22*n^2 - 22*n - 35)*x^2 + (2*n^4 - 4*n^3 - 17*n^2 + 19*n + 21)*x^3 + (n^4 - 2*n^3 - 3*n^2 + 4*n + 4)*x^4)*deriv(v[i - 1], x))/(9*n - 9 - (11*n^3 - 33*n^2 + 28*n - 6)*x + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^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 shift_x_plus_one of chebyshev_t?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from chebyshev_t?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] 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
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023