A299989 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	1	4	3																											8
2	1	8	22	24	9																									64
3	1	12	57	136	171	108	27																							512
4	1	16	108	400	886	1200	972	432	81																					4096
5	1	20	175	880	2810	5944	8430	7920	4725	1620	243																			32768
6	1	24	258	1640	6855	19824	40636	59472	61695	44280	20898	5832	729																	262144
7	1	28	357	2744	14189	52164	140497	281488	421491	469476	383103	222264	86751	20412	2187															2097152
8	1	32	472	4256	26236	117152	391720	999968	1968934	2999904	3525480	3163104	2125116	1034208	344088	69984	6561													16777216
9	1	36	603	6240	44676	234864	939036	2918304	7143966	13875544	21431898	26264736	25353972	19023984	10856268	4548960	1318761	236196	19683											134217728
10	1	40	750	8760	71445	432288	2012520	7379040	21634290	51206320	98365972	153618960	194708610	199234080	163014120	105045984	52083405	19158120	4920750	787320	59049									1073741824
11	1	44	913	11880	108735	744348	3956007	16725984	57188010	159880600	368094122	700701808	1104282366	1438925400	1544076270	1354804704	961309701	542629692	237803445	77944680	17970579	2598156	177147							8589934592
12	1	48	1092	15664	158994	1214928	7259604	34783056	135959967	438810592	1179180552	2652720096	5011371964	7958160288	10612624968	11847885984	11012757327	8452282608	5292251316	2657047536	1043159634	308314512	64481508	8503056	531441					68719476736
13	1	52	1287	20176	224926	1897896	12596298	67466256	296871003	1086999628	3342302821	8685874080	19159794004	35961808432	57479382012	78172866720	90242176167	88046969868	72139653729	49182900624	27548103726	12452095656	4427218458	1191372624	227988189	27634932	1594323			549755813888
14	1	56	1498	25480	309491	2858128	20862660	123545136	604524921	2476882408	8580914342	25316084248	63930198787	138658606688	258805997752	415975820064	575371789083	683534274696	695054061702	601882425144	440698667409	270193212432	136879912260	56256533424	18275134059	4513705560	796098618	89282088	4782969	4398046511104

Technical details

Node id: n-45c3e3fbc36932e1 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-45b1586921223be0

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 1,  4,   3;
  2: 1,  8,  22,   24,     9;
  3: 1, 12,  57,  136,   171,    108,     27;
  4: 1, 16, 108,  400,   886,   1200,    972,     432,      81;
  5: 1, 20, 175,  880,  2810,   5944,   8430,    7920,    4725,     1620,      243;
  6: 1, 24, 258, 1640,  6855,  19824,  40636,   59472,   61695,    44280,    20898,     5832,      729;
  7: 1, 28, 357, 2744, 14189,  52164, 140497,  281488,  421491,   469476,   383103,   222264,    86751,    20412,     2187;
  8: 1, 32, 472, 4256, 26236, 117152, 391720,  999968, 1968934,  2999904,  3525480,  3163104,  2125116,  1034208,   344088,   69984,    6561;
  9: 1, 36, 603, 6240, 44676, 234864, 939036, 2918304, 7143966, 13875544, 21431898, 26264736, 25353972, 19023984, 10856268, 4548960, 1318761, 236196, 19683;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x + 3*x^2)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + 4*x + 3*x^2)*v[i - 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) f…?
  - Parent-row provers of polytool polytool ? 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 ?

A299989: Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(…

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.