A147644 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(-6 - 12\,x + 2\,n + 4\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 4\,x\)

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

\(P_{3}(x) = 12 + 72\,x + 128\,x^{2} + 64\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	4	16	16													36
3	12	72	128	64												276
4	24	192	544	640	256											1656
5	48	480	1856	3456	3072	1024										9936
6	96	1152	5632	14336	19968	14336	4096									59616
7	192	2688	15872	51200	97280	108544	65536	16384								357696
8	384	6144	42496	165888	399360	606208	565248	294912	65536							2146176
9	768	13824	109568	501760	1462272	2809856	3555328	2850816	1310720	262144						12877056
10	1536	30720	274432	1441792	4931584	11468800	18350080	19922944	14024704	5767168	1048576					77262336
11	3072	67584	671744	3981312	15630336	42663936	82575360	113246208	107741184	67633152	25165824	4194304				463574016
12	6144	147456	1613824	10649600	47185920	147849216	335806464	556793856	668467200	566231040	320864256	109051904	16777216			2781444096
13	12288	319488	3817472	27754496	136970240	484442112	1263009792	2456813568	3564109824	3806330880	2906652672	1501560832	469762048	67108864		16688664576
14	24576	688128	8912896	70778880	384958464	1516765184	4463788032	9965666304	16955473920	21869101056	21038628864	14629732352	6945767424	2013265920	268435456	100131987456

Technical details

Node id: n-59281eca5ea2cc86 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-32ce21d4b68ee7fe

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:   1;
  1:   2,     4;
  2:   4,    16,     16;
  3:  12,    72,    128,     64;
  4:  24,   192,    544,    640,     256;
  5:  48,   480,   1856,   3456,    3072,    1024;
  6:  96,  1152,   5632,  14336,   19968,   14336,    4096;
  7: 192,  2688,  15872,  51200,   97280,  108544,   65536,   16384;
  8: 384,  6144,  42496, 165888,  399360,  606208,  565248,  294912,   65536;
  9: 768, 13824, 109568, 501760, 1462272, 2809856, 3555328, 2850816, 1310720, 262144;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (2*n-6+(4*n-12)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 4 + 16*x + 16*x^2, P(3,x) = 12 + 72*x + 128*x^2 + 64*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 4 + 16*x + 16*x^2;
P[3, x_] := 12 + 72*x + 128*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 6 + (4*n - 12)*x)*P[n - 1, x])/(n - 3)]];
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] = 2 + 4*x; v[3] = 4 + 16*x + 16*x^2; v[4] = 12 + 72*x + 128*x^2 + 64*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((2*n - 6 + (4*n - 12)*x)*v[i - 1])/(n - 3)); 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 ≥ 4
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n, k) = binomial(n, k) if n < 3, otherwise T(n, k) = binomial(n, k) + 2*binomial(n-2, k-1).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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.