A290315 → D

nonnegative coefficients

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

Recurrence

\(\left(-11 + 4\,n\right)\,P_{n}(x) = \left(21 - 364\,x + 4004\,x^{2} - 12\,n + 320\,n x - 4976\,n x^{2} - 64\,n^{2} x + 1984\,n^{2} x^{2} - 256\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(14\,x - 336\,x^{2} + 616\,x^{3} - 8\,n x + 304\,n x^{2} - 576\,n x^{3} - 64\,n^{2} x^{2} + 128\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-40 + 396\,x + 16\,n - 320\,n x + 64\,n^{2} x\right)\,P_{n-1}(x) + \left(-22\,x + 44\,x^{2} + 8\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	2														2
3	16	24													40
4	66	568	360												994
5	224	5744	17856	6720											30544
6	706	42160	349008	584256	151200										1127330
7	2160	264456	4593216	19218624	20651520	3991680									48721656
8	6530	1521000	48813240	405690560	1043958240	795346560	121080960								2416417090
9	19648	8310240	455767680	6645093760	32782295040	57944001024	33364961280	4151347200							135351795872
10	59010	43987552	3913364640	92621116800	775923605440	2562854699520	3339730469376	1519565506560	158789030400						8453441839298
11	177104	228199160	31748320128	1157094861440	15250190627840	83596960726272	199595814512640	201407015430144	74816461762560	6704425728000					582559940345288
12	531394	1168624024	247568347656	13377030298240	263212422975040	2221374672152832	8650664893500672	15741832386521088	12750657391434240	3964745430374400	309744468633600				43915857433393186
13	1594272	5934519888	1876719188544	146140380840384	4132711955312640	50987714267890176	300805823158278144	880572548738906112	1269417053657751552	848299097523916800	225185121449164800	15543540607795200			3595091634395158512
14	4782914	29972322576	13938194504688	1530460561183168	60439220544578400	1049662124499835392	8914026679168690688	39034199181223655424	89501435763361058304	105263011411091804160	59300740215356682240	13654039306018406400	841941782922240000		317621040112919744354
15	14348848	150837327336	101973292951488	15521065428569408	837008239123468800	19882935178698350976	234128076764681279488	1458203192909553997824	4950124361340729421824	9171746021227676866560	9006011653398468673536	4352237411761857576960	880661231200390348800	48962152914554880000	30122809668125308061848

Technical details

Node id: n-cc6f0733a6e78a5c · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-2fd84d8d5849b0ab

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     0;
   2:     2;
   3:    16,       24;
   4:    66,      568,        360;
   5:   224,     5744,      17856,        6720;
   6:   706,    42160,     349008,      584256,       151200;
   7:  2160,   264456,    4593216,    19218624,     20651520,       3991680;
   8:  6530,  1521000,   48813240,   405690560,   1043958240,     795346560,     121080960;
   9: 19648,  8310240,  455767680,  6645093760,  32782295040,   57944001024,   33364961280,    4151347200;
  10: 59010, 43987552, 3913364640, 92621116800, 775923605440, 2562854699520, 3339730469376, 1519565506560, 158789030400;

%F Formula

The row polynomials satisfy (4*n-11)*P(n,x) = -(12*n-21+(64*n^2-320*n+364)*x+(256*n^3-1984*n^2+4976*n-4004)*x^2)*P(n-2,x) - ((8*n-14)*x+(64*n^2-304*n+336)*x^2-(128*n^2-576*n+616)*x^3)*d/dx P(n-2,x) + (16*n-40+(64*n^2-320*n+396)*x)*P(n-1,x) + ((8*n-22)*x-(16*n-44)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 21 + (64*n^2 - 320*n + 364)*x + (256*n^3 - 1984*n^2 + 4976*n - 4004)*x^2)*P[n - 2, x] - ((8*n - 14)*x + (64*n^2 - 304*n + 336)*x^2 - (128*n^2 - 576*n + 616)*x^3)*D[P[n - 2, x], x] + (16*n - 40 + (64*n^2 - 320*n + 396)*x)*P[n - 1, x] + ((8*n - 22)*x - (16*n - 44)*x^2)*D[P[n - 1, x], x])/(4*n - 11)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = (-(12*n - 21 + (64*n^2 - 320*n + 364)*x + (256*n^3 - 1984*n^2 + 4976*n - 4004)*x^2)*v[i - 2] - ((8*n - 14)*x + (64*n^2 - 304*n + 336)*x^2 - (128*n^2 - 576*n + 616)*x^3)*deriv(v[i - 2], x) + (16*n - 40 + (64*n^2 - 320*n + 396)*x)*v[i - 1] + ((8*n - 22)*x - (16*n - 44)*x^2)*deriv(v[i - 1], x))/(4*n - 11)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

References

[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.