A049374 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-36 + 12\,x + 4\,x^{2} - 13\,n + 3\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-20\,x + 20\,x^{2} - 5\,n x\right)\,P_{n-2}'(x) + \left(14 - 3\,x + 2\,n\right)\,P_{n-1}(x) + \left(5\,x\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 7 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	7	1														8
2	60	20	1													81
3	612	348	39	1												1000
4	7224	6120	1140	64	1											14549
5	96264	112824	30780	2820	95	1										242784
6	1419264	2201472	819504	108360	5880	132	1									4554613
7	22796928	45505152	22098384	3937584	304920	10920	175	1								94654064
8	394377984	994194432	610860096	140374080	14503104	734832	18648	224	1							2155063401
9	7285614336	22885099776	17405720640	5003570880	660623040	44307648	1580040	29880	279	1						53286546520
10	142806343680	553176594432	512404776576	180207054720	29505833280	2510500608	117765648	3112560	45540	340	1					1420732027385
11	2955344071680	13997761569792	15595189593984	6597497402496	1310329500480	137307538368	8123102064	280981008	5720220	66660	407	1				40601839547160
12	64320143247360	369770956498944	490610280416256	246386856748032	58362773204736	7369232846976	533928979584	23180084928	615010968	9935640	94380	480	1			1237377977068285
13	1467507047086080	10172482599831552	15943080459930624	9404773713054720	2621733528750336	392246246456064	34061264100864	1802729440512	59791291560	1254461208	16468452	129948	559	1		40037748651002480
14	35006756673208320	290812886710714368	534720560903553024	367310700433797120	119210913973831680	20854439782944768	2134616980624128	134806939787520	5429927543040	141935713920	2412682272	26240760	174720	644	1	1370191256700816285

Technical details

Node id: n-4019cc30eb72d880 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-2f94d007fcf36594

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:          7,           1;
  2:         60,          20,           1;
  3:        612,         348,          39,          1;
  4:       7224,        6120,        1140,         64,         1;
  5:      96264,      112824,       30780,       2820,        95,        1;
  6:    1419264,     2201472,      819504,     108360,      5880,      132,       1;
  7:   22796928,    45505152,    22098384,    3937584,    304920,    10920,     175,     1;
  8:  394377984,   994194432,   610860096,  140374080,  14503104,   734832,   18648,   224,   1;
  9: 7285614336, 22885099776, 17405720640, 5003570880, 660623040, 44307648, 1580040, 29880, 279, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+13*n+36-(3*n+12)*x-4*x^2)*P(n-2,x) - ((5*n+20)*x-20*x^2)*d/dx P(n-2,x) + (2*n+14-3*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 7 + x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 13*n + 36 - (3*n + 12)*x - 4*x^2)*P[n - 2, x] - ((5*n + 20)*x - 20*x^2)*D[P[n - 2, x], x] + (2*n + 14 - 3*x)*P[n - 1, x] + 5*x*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] = 1; v[2] = 7 + x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 13*n + 36 - (3*n + 12)*x - 4*x^2)*v[i - 2] - ((5*n + 20)*x - 20*x^2)*deriv(v[i - 2], x) + (2*n + 14 - 3*x)*v[i - 1] + 5*x*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 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 A triangle of numbers related to triangle A030527.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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.