A272867 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6\,n + 18\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 18 + 18\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	18	18													36
2	216	864	972	324											2376
3	1944	13608	34992	40824	21870	4374									117612
4	15552	155520	629856	1329696	1574640	1049760	367416	52488							5174928
5	116640	1516320	8398080	25894080	48813840	58261680	44089920	20470320	5314410	590490					213465780
6	839808	13436928	94478400	384212160	1001471040	1753519104	2103089184	1728954720	956593800	340122240	70150212	6377292			8453244888
7	5878656	111694464	952342272	4814619264	16070775840	37339753248	62060971392	74679506496	65086642152	40623350040	17677853424	5089079016	870500358	66961566	325449928188

All 15 rows

Technical details

Node id: n-22d01599550a0f21 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-b643eea373acac66

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:        18,         18;
  2:       216,        864,         972,          324;
  3:      1944,      13608,       34992,        40824,         21870,          4374;
  4:     15552,     155520,      629856,      1329696,       1574640,       1049760,         367416,          52488;
  5:    116640,    1516320,     8398080,     25894080,      48813840,      58261680,       44089920,       20470320,        5314410,         590490;
  6:    839808,   13436928,    94478400,    384212160,    1001471040,    1753519104,     2103089184,     1728954720,      956593800,      340122240,       70150212,        6377292;
  7:   5878656,  111694464,   952342272,   4814619264,   16070775840,   37339753248,    62060971392,    74679506496,    65086642152,    40623350040,    17677853424,     5089079016,      870500358,       66961566;
  8:  40310784,  886837248,  8888527872,  53754430464,  219038722560,  636164637696,  1358992422144,  2172836918016,  2620913956992,  2385617391360,  1626362515008,   816395412672,   292488120288,    70711413696,   10331213040,    688747536;
  9: 272097792, 6802444800, 78364164096, 551814322176, 2657034938880, 9273909044736, 24273299828736, 48627412701696, 75450915073728, 91176398815680, 85823452965888, 62598886051968, 34973222383008, 14666190031584, 4463084033280, 929809173600, 118550669634, 6973568802;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (6*n+18*n*x+9*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 18 + 18*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 18 + 18*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 18*n*x + 9*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] = 18 + 18*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 18*n*x + 9*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 basis_map_A013610 of 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…?
  - Geometric basis map (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 [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.