A143491 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 48\,x + 144\,x^{2} - 6\,n - 16\,n x - n^{2}\right)\,P_{n-2}(x) + \left(48\,x^{2} + 64\,x^{3} + 16\,n x^{2}\right)\,P_{n-2}'(x) + \left(7 + 16\,x + 2\,n\right)\,P_{n-1}(x) + \left(-16\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 12 + 112\,x + 144\,x^{2}\)

\(P_{1}(x) = 60 + 752\,x + 1728\,x^{2} + 1024\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	12	112	144								268
1	60	752	1728	1024							3564
2	360	5472	17136	18432	6400						47800
3	2520	44064	169200	250880	160000	36864					663528
4	20160	392832	1750176	3210240	2848000	1216512	200704				9638624
5	181440	3858048	19287072	41337856	45696000	27353088	8429568	1048576			147191648
6	1814400	41483520	227593152	550531072	715321600	536739840	233218048	54525952	5308416		2366536000
7	19958400	485349120	2876876352	7674281984	11309356800	10024390656	5487648768	1818230784	334430208	26214400	40056737472

All 15 rows

Technical details

Node id: n-9196ed29b60d5d3c · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-dcd95a62876f0c72

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:         12,         112,          144;
  1:         60,         752,         1728,          1024;
  2:        360,        5472,        17136,         18432,          6400;
  3:       2520,       44064,       169200,        250880,        160000,         36864;
  4:      20160,      392832,      1750176,       3210240,       2848000,       1216512,        200704;
  5:     181440,     3858048,     19287072,      41337856,      45696000,      27353088,       8429568,       1048576;
  6:    1814400,    41483520,    227593152,     550531072,     715321600,     536739840,     233218048,      54525952,      5308416;
  7:   19958400,   485349120,   2876876352,    7674281984,   11309356800,   10024390656,    5487648768,    1818230784,    334430208,    26214400;
  8:  239500800,  6143523840,  38890658304,  112549171200,  183676544000,  185434583040,  120429023232,   50488934400,  13217955840,  1966080000,   126877696;
  9: 3113510400, 83697822720, 560870272512, 1739695017984, 3091227392000, 3468626472960, 2575165587456, 1285536350208, 427433656320, 90832896000, 11165237248, 603979776;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(16*n+48)*x-144*x^2)*P(n-2,x) + ((16*n+48)*x^2+64*x^3)*d/dx P(n-2,x) + (2*n+7+16*x)*P(n-1,x) - 16*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 12 + 112*x + 144*x^2, P(1,x) = 60 + 752*x + 1728*x^2 + 1024*x^3.

%t Mathematica

P[0, x_] := 12 + 112*x + 144*x^2;
P[1, x_] := 60 + 752*x + 1728*x^2 + 1024*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (16*n + 48)*x - 144*x^2)*P[n - 2, x] + ((16*n + 48)*x^2 + 64*x^3)*D[P[n - 2, x], x] + (2*n + 7 + 16*x)*P[n - 1, x] - 16*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 12 + 112*x + 144*x^2; v[2] = 60 + 752*x + 1728*x^2 + 1024*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (16*n + 48)*x - 144*x^2)*v[i - 2] + ((16*n + 48)*x^2 + 64*x^3)*deriv(v[i - 2], x) + (2*n + 7 + 16*x)*v[i - 1] - 16*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 - 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 basis_map_A013611 of Unsigned 2-Stirling numbers of the first kind.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Unsigned 2-Stirling numbers of the first kind.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Unsigned 2-Stirling numbers of the first kind.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

A136124 → basis A013611 → (θ+1)²

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.