A272867 → D

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	P_n(1)
0																													0
1	4	2																											6
2	8	36	24	4																									72
3	12	102	264	204	60	6																							648
4	16	200	912	1816	1520	600	112	8																					5184
5	20	330	2160	7080	12120	10620	5040	1320	180	10																			38880
6	24	492	4200	19260	51120	79416	71568	38520	12600	2460	264	12																	279936
7	28	686	7224	42644	154420	353682	513744	471576	277956	106610	26488	4116	364	14															1959552
8	32	912	11424	82544	379680	1158864	2380448	3291696	3060576	1931440	835296	247632	49504	6384	480	16													13436928
9	36	1170	16992	145296	811440	3105144	8320032	15718896	20933784	19648620	13074336	6210288	2109744	508536	84960	9360	612	18											90699264
10	40	1460	24120	238260	1566240	7218000	23946720	57893520	102366000	132336280	125114000	86840280	44472480	16842000	4698720	953040	136680	13140	760	20									604661760
11	44	1782	33000	369820	2797740	15093342	59823456	176988240	393675480	659663180	832507632	791595816	568642360	309729420	128193120	40248912	9512316	1664190	209000	17820	924	22							3990767616
12	48	2136	43824	549384	4701840	29077224	134175888	470591352	1267038432	2630567280	4216184544	5215952208	4982763552	3682794192	2111730720	941182704	325855728	87231672	17866992	2746920	306768	23496	1104	24					26121388032
13	52	2522	56784	787384	7521800	52470132	276452176	1122736472	3557211372	8850088390	17339283104	26770529136	32567986672	31232283992	23644476960	14160141424	6719177036	2526157062	750370192	174900440	31591560	4330612	435344	30264	1300	26			169789022208
14	56	2940	72072	1095276	11553360	89755848	531843312	2456478024	8964964008	26063337300	60627374808	113052961476	169065492960	202757433648	195075568800	150737281968	93696851976	46914007140	18926035128	6141195060	1595529936	329104776	53145456	6571656	600600	38220	1512	28	1097098297344

Technical details

Node id: n-749edefcbd70f36b · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-70b67aafe0c2287c

Observed on the stored rows ?

Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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:  4,    2;
  2:  8,   36,    24,      4;
  3: 12,  102,   264,    204,     60,       6;
  4: 16,  200,   912,   1816,   1520,     600,     112,        8;
  5: 20,  330,  2160,   7080,  12120,   10620,    5040,     1320,      180,       10;
  6: 24,  492,  4200,  19260,  51120,   79416,   71568,    38520,    12600,     2460,      264,      12;
  7: 28,  686,  7224,  42644, 154420,  353682,  513744,   471576,   277956,   106610,    26488,    4116,     364,     14;
  8: 32,  912, 11424,  82544, 379680, 1158864, 2380448,  3291696,  3060576,  1931440,   835296,  247632,   49504,   6384,   480,   16;
  9: 36, 1170, 16992, 145296, 811440, 3105144, 8320032, 15718896, 20933784, 19648620, 13074336, 6210288, 2109744, 508536, 84960, 9360, 612, 18;

%F Formula

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

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4*n*x + n*x^2)*P[n - 1, x])/(n - 1)]];
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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + 4*n*x + n*x^2)*v[i - 1])/(n - 1)); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
  - ↑ Parent-row provers of polytool (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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.