A156319 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 3 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	3														6
2	3	12	9													24
3	3	21	45	27												96
4	3	30	108	162	81											384
5	3	39	198	486	567	243										1536
6	3	48	315	1080	2025	1944	729									6144
7	3	57	459	2025	5265	8019	6561	2187								24576
8	3	66	630	3402	11340	23814	30618	21870	6561							98304
9	3	75	828	5292	21546	57834	102060	113724	72171	19683						393216
10	3	84	1053	7776	37422	122472	275562	419904	413343	236196	59049					1572864
11	3	93	1305	10935	60750	234738	642978	1246590	1673055	1476225	767637	177147				6291456
12	3	102	1584	14850	93555	416988	1347192	3175524	5412825	6495390	5196312	2480058	531441			25165824
13	3	111	1890	19602	138105	697653	2598156	7217100	14939397	22733865	24682482	18068994	7971615	1594323		100663296
14	3	120	2223	25272	196911	1111968	4691115	15011568	36590697	67552056	92884077	92116440	62178597	25509168	4782969	402653184

Technical details

Node id: n-5c4750b14b7bcd34 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-fb13869215ab1dd7

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: 1;
  1: 3,  3;
  2: 3, 12,   9;
  3: 3, 21,  45,   27;
  4: 3, 30, 108,  162,    81;
  5: 3, 39, 198,  486,   567,   243;
  6: 3, 48, 315, 1080,  2025,  1944,    729;
  7: 3, 57, 459, 2025,  5265,  8019,   6561,   2187;
  8: 3, 66, 630, 3402, 11340, 23814,  30618,  21870,  6561;
  9: 3, 75, 828, 5292, 21546, 57834, 102060, 113724, 72171, 19683;

%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) = 1, P(1,x) = 3 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 3*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] = 1; v[2] = 3 + 3*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: (1, 2, 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

Related ?

Same fingerprint ?

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.