A093563 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 7 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	7	3														10
2	14	27	9													50
3	28	96	99	27												250
4	56	276	486	351	81											1250
5	112	720	1800	2160	1215	243										6250
6	224	1776	5760	9720	8910	4131	729									31250
7	448	4224	16848	36720	46980	34992	13851	2187								156250
8	896	9792	46368	123984	204120	210924	132678	45927	6561							781250
9	1792	22272	122112	387072	780192	1034208	898128	489888	150903	19683						3906250
10	3584	49920	311040	1140480	2721600	4408992	4898880	3674160	1771470	492075	59049					19531250
11	7168	110592	771840	3214080	8864640	16982784	23024736	22044960	14565420	6298560	1594323	177147				97656250
12	14336	242688	1875456	8743680	27371520	60559488	96997824	113164128	95265720	56293380	22084326	5137263	531441			488281250
13	28672	528384	4478976	23113728	80974080	203233536	375674112	517321728	530023824	398383920	213048792	76527504	16474671	1594323		2441406250
14	57344	1142784	10543104	59664384	231289344	649389312	1361048832	2161665792	2612012832	2386839312	1621249344	792201384	262531854	52612659	4782969	12207031250

Technical details

Node id: n-b707c41a3afefcce · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-b305c23e270077b9

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:    7,     3;
  2:   14,    27,      9;
  3:   28,    96,     99,     27;
  4:   56,   276,    486,    351,     81;
  5:  112,   720,   1800,   2160,   1215,     243;
  6:  224,  1776,   5760,   9720,   8910,    4131,    729;
  7:  448,  4224,  16848,  36720,  46980,   34992,  13851,   2187;
  8:  896,  9792,  46368, 123984, 204120,  210924, 132678,  45927,   6561;
  9: 1792, 22272, 122112, 387072, 780192, 1034208, 898128, 489888, 150903, 19683;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(3*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 7 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (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] = 7 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from (6,1)-Pascal triangle.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - 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
[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.