A128813 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	2														6
2	28	44	12													84
3	292	884	756	144												2076
4	4332	20284	32476	19584	2880											79556
5	87792	565984	1376656	1529544	720000	86400										4366376
6	2345376	19412224	64306768	107326368	92176848	35769600	3628800									324965984
7	80158496	813494016	3435923856	7752722336	9926204688	6983042688	2311545600	203212800								31506304480
8	3418609568	41189356800	212327203152	607823932832	1046617911504	1091766198528	652318292736	188581478400	14631321600							3858674305120
9	178195506368	2489120668800	15184246815552	52869114168992	115074675129024	160810760125728	141909775151616	73821175856640	18962192793600	1316818944000						582616075160320
10	11156211680768	177416505529600	1251810610834752	5145130087879232	13585136869285504	23930189183377728	28206566232601536	21606866836992000	9964402593408000	2304433152000000	144850083840000					106327958367429120
11	826578277995008	14753033658822400	118303025097550272	561371471580920000	1746298445976496000	3723820934554269888	5520545706404093760	5635404813080362752	3818174950014336000	1582681228909056000	333010342748160000	19120211066880000				23074310741368942080
12	71563923514248704	1417117842228769024	12736807872963624000	68592243595356183872	245977857834111168640	616999656709518479424	1105459313343567562944	1416799966639795937280	1276849633720810125312	776172068216395776000	292345420064864256000	56442863069429760000	2982752926433280000			5872847265758989171200
13	7161235042079483904	155846755565464675328	1552435855240695021056	9360373760064473123328	38051751259578678185984	109897932567348454296704	231317518264885219339008	357806659261159759280256	404276526059091595112448	326398671114079547510784	180025005952791576576000	62157878541498765312000	11089875380478935040000	542861032610856960000		1732640497039436099916800
14	820379270514323165184	19466186451558740953088	212928643157243648086016	1421667450089901292290048	6464719670051633785889024	21145111060168212149863424	51193806674491912290067968	92930055422502400891094016	126581901174563505918605568	127975093999940887852621824	93632867219404360507760640	47284782338724598628352000	15075544494918444687360000	2500417916205607157760000	114000816848279961600000	586553183446789061835468800

Technical details

Node id: n-78fe0e73e916e68a · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-6aa6dbddb3aeaca3

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:            4,             2;
  2:           28,            44,             12;
  3:          292,           884,            756,            144;
  4:         4332,         20284,          32476,          19584,            2880;
  5:        87792,        565984,        1376656,        1529544,          720000,           86400;
  6:      2345376,      19412224,       64306768,      107326368,        92176848,        35769600,         3628800;
  7:     80158496,     813494016,     3435923856,     7752722336,      9926204688,      6983042688,      2311545600,      203212800;
  8:   3418609568,   41189356800,   212327203152,   607823932832,   1046617911504,   1091766198528,    652318292736,   188581478400,    14631321600;
  9: 178195506368, 2489120668800, 15184246815552, 52869114168992, 115074675129024, 160810760125728, 141909775151616, 73821175856640, 18962192793600, 1316818944000;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^2)*P[n - 2, x] + (4*n^3 + 4*n + (8*n^3 - 8*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] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^2)*v[i - 2] + (4*n^3 + 4*n + (8*n^3 - 8*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 basis_map_A013609 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
  - Geometric basis map (elementary) 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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

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.