A167194 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 - 8\,x + n + 4\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 4\,x\)

\(P_{2}(x) = 4 + 16\,x + 16\,x^{2}\)

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	4														7
2	4	16	16													36
3	4	32	80	64												180
4	4	48	208	384	256											900
5	4	64	400	1216	1792	1024										4500
6	4	80	656	2816	6656	8192	4096									22500
7	4	96	976	5440	17920	34816	36864	16384								112500
8	4	112	1360	9344	39680	106496	176128	163840	65536							562500
9	4	128	1808	14784	77056	265216	602112	868352	720896	262144						2812500
10	4	144	2320	22016	136192	573440	1662976	3276800	4194304	3145728	1048576					14062500
11	4	160	2896	31296	224256	1118208	3956736	9928704	17301504	19922944	13631488	4194304				70312500
12	4	176	3536	42880	349440	2015232	8429568	25755648	57016320	89128960	93323264	58720256	16777216			351562500
13	4	192	4240	57024	520960	3412992	16490496	59473920	160038912	317194240	449839104	432013312	251658240	67108864		1757812500
14	4	208	5008	73984	749056	5496832	30142464	125435904	397934592	957349888	1718616064	2231369728	1979711488	1073741824	268435456	8789062500

Technical details

Node id: n-518a0da3e339cb15 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-7d31072f3aba30ae

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,   4;
  2: 4,  16,   16;
  3: 4,  32,   80,    64;
  4: 4,  48,  208,   384,   256;
  5: 4,  64,  400,  1216,  1792,   1024;
  6: 4,  80,  656,  2816,  6656,   8192,   4096;
  7: 4,  96,  976,  5440, 17920,  34816,  36864,  16384;
  8: 4, 112, 1360,  9344, 39680, 106496, 176128, 163840,  65536;
  9: 4, 128, 1808, 14784, 77056, 265216, 602112, 868352, 720896, 262144;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(4*n-8)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 4*x, P(2,x) = 4 + 16*x + 16*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[2, x_] := 4 + 16*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (4*n - 8)*x)*P[n - 1, x])/(n - 2)]];
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 + 4*x; v[3] = 4 + 16*x + 16*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (4*n - 8)*x)*v[i - 1])/(n - 2)); 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 ≥ 3
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows. A130713 in the columns.?
  - 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.