A128813 → D

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-4 - 4\,n + 4\,n x - n^{2} x^{2} - 4\,n^{3} x + n^{3} x^{2} + n^{4} x^{2} - n^{5} x^{2}\right)\,P_{n-2}(x) + \left(8\,n - 4\,n x + 4\,n^{3} x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	4	6													10
3	10	54	54												118
4	20	254	864	720											1858
5	35	854	6579	18000	13500										38968
6	56	2324	33480	202212	486000	340200									1064272
7	84	5460	131088	1452132	7563420	16669800	11113200								36935184
8	120	11508	425928	7744356	72909360	343409544	711244800	457228800							1592974416
9	165	22308	1202718	33300036	508529385	4280514984	18740245860	37035532800	23147208000						83746556256
10	220	40458	3043128	121499356	2797906860	37843454394	293406624000	1214993844000	2314720800000	1414551600000					5278438812416
11	286	69498	7048470	389294620	12821603730	259437677706	3207352612338	23346236340000	92528013717000	171160743600000	102696446160000				393211448123648
12	364	114114	15179736	1122335500	50777829180	1459539786834	26816181283584	309258812068416	2141160252552000	8190255265740000	14788288247040000	8738533964160000			34195824178089728
13	455	180362	30756297	2964143468	178443492305	7004478733290	181770655319127	3098141665561152	33801531138056088	224685347468220000	834632840347614000	1476812239943040000	861473806633440000		3434692863768556544
14	560	275912	59163312	7270025048	567487322480	29488358763720	1039819300147152	24910620303856392	399769515382467168	4168197313574763600	26785790472927024000	97080210316124280000	168848866100154240000	97412807365473600000	394721621586415929344

Technical details

Node id: n-b0cc633b347bd346 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-e6a4ddaeab3da9d0

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 -1)
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:   0;
  1:   1;
  2:   4,     6;
  3:  10,    54,      54;
  4:  20,   254,     864,      720;
  5:  35,   854,    6579,    18000,     13500;
  6:  56,  2324,   33480,   202212,    486000,     340200;
  7:  84,  5460,  131088,  1452132,   7563420,   16669800,    11113200;
  8: 120, 11508,  425928,  7744356,  72909360,  343409544,   711244800,   457228800;
  9: 165, 22308, 1202718, 33300036, 508529385, 4280514984, 18740245860, 37035532800, 23147208000;

%F Formula

The row polynomials satisfy (4*n-4)*P(n,x) = -(4*n+4+(4*n^3-4*n)*x+(n^5-n^4-n^3+n^2)*x^2)*P(n-2,x) + (8*n+(4*n^3-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n + 4 + (4*n^3 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*P[n - 2, x] + (8*n + (4*n^3 - 4*n)*x)*P[n - 1, x])/(4*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n + 4 + (4*n^3 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*v[i - 2] + (8*n + (4*n^3 - 4*n)*x)*v[i - 1])/(4*n - 4)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
  - 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

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.