A136157 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-1 - 3\,x + n + 3\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 3\)

\(P_{1}(x) = 4 + 9\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	3															3
1	4	9														13
2	4	21	27													52
3	4	33	90	81												208
4	4	45	189	351	243											832
5	4	57	324	918	1296	729										3328
6	4	69	495	1890	4050	4617	2187									13312
7	4	81	702	3375	9720	16767	16038	6561								53248
8	4	93	945	5481	19845	45927	66339	54675	19683							212992
9	4	105	1224	8316	36288	105462	204120	253692	183708	59049						851968
10	4	117	1539	11988	61236	214326	520506	866052	944784	610173	177147					3407872
11	4	129	1890	16605	97200	398034	1163484	2427570	3542940	3444525	2007666	531441				13631488
12	4	141	2277	22275	147015	689634	2357586	5918022	10825650	14073345	12341241	6554439	1594323			54525952
13	4	153	2700	29106	213840	1130679	4426488	12990780	28579716	46550295	54561276	43578162	21257640	4782969		218103808
14	4	165	3159	37206	301158	1772199	7818525	26270244	67552056	132289443	194212161	207261990	151992126	68555889	14348907	872415232

Technical details

Node id: n-33acd11b57e0c597 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-977fd250db0d872d

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 3;
  1: 4,   9;
  2: 4,  21,   27;
  3: 4,  33,   90,   81;
  4: 4,  45,  189,  351,   243;
  5: 4,  57,  324,  918,  1296,    729;
  6: 4,  69,  495, 1890,  4050,   4617,   2187;
  7: 4,  81,  702, 3375,  9720,  16767,  16038,   6561;
  8: 4,  93,  945, 5481, 19845,  45927,  66339,  54675,  19683;
  9: 4, 105, 1224, 8316, 36288, 105462, 204120, 253692, 183708, 59049;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (n-1+(3*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 3, P(1,x) = 4 + 9*x.

%t Mathematica

P[0, x_] := 3;
P[1, x_] := 4 + 9*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (3*n - 3)*x)*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] = 3; v[2] = 4 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (3*n - 3)*x)*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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle by columns, (3, 1, 0, 0, 0, ...) in every column.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.