A257618 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 4\,n\right)\,P_{n}(x) = \left(12 - 168\,x - 84\,x^{2} + 48\,n - 576\,n x - 176\,n x^{2} + 384\,n^{2} x + 576\,n^{2} x^{2} - 256\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(48\,x - 48\,x^{3} + 192\,n x - 64\,n x^{2} - 128\,n x^{3} - 256\,n^{2} x^{2} + 256\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-12 + 36\,x - 16\,n - 96\,n x + 64\,n^{2} x\right)\,P_{n-1}(x) + \left(-24\,x + 24\,x^{2} + 32\,n x - 32\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	4	80	12													96
3	8	944	1416	32												2400
4	16	9856	50976	19712	80											80640
5	32	99648	1301952	1735936	249120	192										3386880
6	64	999168	28517184	90268672	47528640	2997504	448									170311680
7	128	9998080	576256896	3639764480	4549705600	1152513792	34993280	1024								9963233280
8	256	99995648	11122480128	126904262656	309379980800	190356393984	25952453632	399982592	2304							664215552000
9	512	999990272	208904263680	4041009504256	17181393136640	20617671763968	7071766632448	557078036480	4499956224	5120						49683323289600
10	1024	9999978496	3859275783168	121221510168576	836730460661760	1731884428173312	1171422644926464	242443020337152	11577827349504	49999892480	11264					4119199167283200
11	2048	99999952896	70576961710080	3491375533301760	37237395149721600	122942973622984704	143433469226815488	59579832239554560	7855594949928960	235256539033600	549999740928	24576				374847124222771200
12	4096	999999897600	1282685304987648	97739357421240320	1554109916587929600	7755327594585980928	14343346922681548800	10340436792781307904	2797397849858273280	244348393553100800	4703179451621376	5999999385600	53248			37138699692225331200
13	8192	9999999778816	23223335475953664	2681463552964231168	61880627725454336000	448809264366375223296	1241944513367728816128	1419365158134547218432	673213896549562834944	123761255450908672000	7374024770651635712	92893341903814656	64999998562304	114688		3979146395595571200000
14	16384	99999999524864	419490038534651904	72504852634184450048	2378791356844281118720	24344988845408104415232	96655539024597333000192	164646358343607477338112	124271407317339428143104	40574981409013507358720	5233340985057418461184	217514557902553350144	1817790166983491584	699999996674048	245760	458397664772609802240000

Technical details

Node id: n-85a3dd1215d7617f · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-b99dd02e1a9529cf

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:   2,         4;
  2:   4,        80,           12;
  3:   8,       944,         1416,            32;
  4:  16,      9856,        50976,         19712,             80;
  5:  32,     99648,      1301952,       1735936,         249120,            192;
  6:  64,    999168,     28517184,      90268672,       47528640,        2997504,           448;
  7: 128,   9998080,    576256896,    3639764480,     4549705600,     1152513792,      34993280,         1024;
  8: 256,  99995648,  11122480128,  126904262656,   309379980800,   190356393984,   25952453632,    399982592,       2304;
  9: 512, 999990272, 208904263680, 4041009504256, 17181393136640, 20617671763968, 7071766632448, 557078036480, 4499956224, 5120;

%F Formula

The row polynomials satisfy (4*n-3)*P(n,x) = (48*n+12+(384*n^2-576*n-168)*x-(256*n^3-576*n^2+176*n+84)*x^2)*P(n-2,x) + ((192*n+48)*x-(256*n^2+64*n)*x^2+(256*n^2-128*n-48)*x^3)*d/dx P(n-2,x) - (16*n+12-(64*n^2-96*n+36)*x)*P(n-1,x) + ((32*n-24)*x-(32*n-24)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((48*n + 12 + (384*n^2 - 576*n - 168)*x - (256*n^3 - 576*n^2 + 176*n + 84)*x^2)*P[n - 2, x] + ((192*n + 48)*x - (256*n^2 + 64*n)*x^2 + (256*n^2 - 128*n - 48)*x^3)*D[P[n - 2, x], x] - (16*n + 12 - (64*n^2 - 96*n + 36)*x)*P[n - 1, x] + ((32*n - 24)*x - (32*n - 24)*x^2)*D[P[n - 1, x], x])/(4*n - 3)]];
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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((48*n + 12 + (384*n^2 - 576*n - 168)*x - (256*n^3 - 576*n^2 + 176*n + 84)*x^2)*v[i - 2] + ((192*n + 48)*x - (256*n^2 + 64*n)*x^2 + (256*n^2 - 128*n - 48)*x^3)*deriv(v[i - 2], x) - (16*n + 12 - (64*n^2 - 96*n + 36)*x)*v[i - 1] + ((32*n - 24)*x - (32*n - 24)*x^2)*deriv(v[i - 1], x))/(4*n - 3)); 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 read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A142459 → 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.