A185410 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-12\,x - 3\,x^{2} - 16\,n x + 2\,n x^{2} + 16\,n^{2} x + 12\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,x - 2\,x^{2} - 2\,x^{3} + 8\,n x - 8\,n x^{2} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 2\,x - 4\,n - 8\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	2														2
2	0	4	3													7
3	0	8	30	4												42
4	0	16	180	144	5											345
5	0	32	888	2064	580	6										3570
6	0	64	3984	20672	17540	2148	7									44415
7	0	128	16992	171840	321200	126720	7602	8								644490
8	0	256	70464	1280256	4502800	3965280	830424	26176	9							10675665
9	0	512	287616	8902912	53625920	88283040	42348208	5111744	88488	10						198648450
10	0	1024	1163520	59169792	573679680	1590692544	1435150416	410350080	30169368	295140	11					4100671575
11	0	2048	4683264	381391872	5698871040	24856328448	37635692640	20506448256	3714216768	172920120	974182	12				92971528650
12	0	4096	18797568	2406993920	53695545600	351142963200	828615959808	760224697344	267055256880	32000914800	970544300	3188496	13			2296134866025
13	0	8192	75331584	14968295424	486730470400	4606818762240	16088393373696	22954011451392	13680664739712	3246569035200	265817929960	5364024912	10362924	14		61349423785650
14	0	16384	301633536	92119941120	4286761518080	57165642347520	284429292602112	597157189582848	554593053312000	225646367230080	37457392047680	2147890922304	29303845116	33480580	15	1763005348479375

Technical details

Node id: n-5b7702ede3a846da · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9988d4525654d117

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: 0,   2;
  2: 0,   4,      3;
  3: 0,   8,     30,       4;
  4: 0,  16,    180,     144,        5;
  5: 0,  32,    888,    2064,      580,        6;
  6: 0,  64,   3984,   20672,    17540,     2148,        7;
  7: 0, 128,  16992,  171840,   321200,   126720,     7602,       8;
  8: 0, 256,  70464, 1280256,  4502800,  3965280,   830424,   26176,     9;
  9: 0, 512, 287616, 8902912, 53625920, 88283040, 42348208, 5111744, 88488, 10;

%F Formula

The row polynomials satisfy (2*n-1)*P(n,x) = ((16*n^2-16*n-12)*x-(8*n^3-12*n^2-2*n+3)*x^2)*P(n-2,x) + ((8*n+4)*x-(8*n^2+8*n+2)*x^2+(8*n^2-2)*x^3)*d/dx P(n-2,x) - (4*n+2-(8*n^2-8*n+2)*x)*P(n-1,x) + ((4*n-2)*x-(4*n-2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(((16*n^2 - 16*n - 12)*x - (8*n^3 - 12*n^2 - 2*n + 3)*x^2)*P[n - 2, x] + ((8*n + 4)*x - (8*n^2 + 8*n + 2)*x^2 + (8*n^2 - 2)*x^3)*D[P[n - 2, x], x] - (4*n + 2 - (8*n^2 - 8*n + 2)*x)*P[n - 1, x] + ((4*n - 2)*x - (4*n - 2)*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (((16*n^2 - 16*n - 12)*x - (8*n^3 - 12*n^2 - 2*n + 3)*x^2)*v[i - 2] + ((8*n + 4)*x - (8*n^2 + 8*n + 2)*x^2 + (8*n^2 - 2)*x^3)*deriv(v[i - 2], x) - (4*n + 2 - (8*n^2 - 8*n + 2)*x)*v[i - 1] + ((4*n - 2)*x - (4*n - 2)*x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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 A triangular decomposition of the double factorial numbers A001147.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A decomposition of the double factorials A001147.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A triangular decomposition of the double factorial numbers A001147.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A decomposition of the double factorials A001147.?
    
    ↑ Liu–Wang theorem (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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.