A200536 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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	2	3	1																											6
2	4	12	13	6	1																									36
3	8	36	66	63	33	9	1																							216
4	16	96	248	360	321	180	62	12	1																					1296
5	32	240	800	1560	1970	1683	985	390	100	15	1																			7776
6	64	576	2352	5760	9420	10836	8989	5418	2355	720	147	18	1																	46656
7	128	1344	6496	19152	38472	55692	59906	48639	29953	13923	4809	1197	203	21	1															279936
8	256	3072	17152	59136	140896	245952	325360	332688	265729	166344	81340	30744	8806	1848	268	24	1													1679616
9	512	6912	43776	172800	476352	973728	1529472	1887408	1854882	1462563	927441	471852	191184	60858	14886	2700	342	27	1											10077696
10	1024	15360	108800	483840	1514880	3549312	6456480	9336960	10901460	10377180	8097453	5188590	2725365	1167120	403530	110916	23670	3780	425	30	1									60466176
11	2048	33792	264704	1309440	4590080	12127104	25075776	41592672	56270280	62795700	58227906	45046719	29113953	15698925	7033785	2599542	783618	189486	35860	5115	517	33	1							362797056
12	4096	73728	632832	3446784	13373184	39333888	91122944	170539776	262394352	335994912	361113192	327572856	251595969	163786428	90278298	41999364	16399647	5329368	1423796	307296	52239	6732	618	36	1					2176782336
13	8192	159744	1490944	8865792	37719552	122234112	313620736	653782272	1127530976	1629712656	1992605472	2074480200	1847023698	1409933619	923511849	518620050	249075684	101857041	35235343	10215348	2450162	477477	73671	8658	728	39	1			13060694016
14	16384	344064	3469312	22364160	103527424	366492672	1031663360	2370660864	4530029504	7295800512	10001879888	11756489472	11910093468	10435418532	7923848253	5217709266	2977523367	1469561184	625117493	227993766	70781711	18520788	4029935	715806	101101	10920	847	42	1	78364164096

Technical details

Node id: n-22db44d78c07f843 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-99cef2a9f1688781

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:   2,    3,     1;
  2:   4,   12,    13,      6,      1;
  3:   8,   36,    66,     63,     33,      9,       1;
  4:  16,   96,   248,    360,    321,    180,      62,      12,       1;
  5:  32,  240,   800,   1560,   1970,   1683,     985,     390,     100,      15,      1;
  6:  64,  576,  2352,   5760,   9420,  10836,    8989,    5418,    2355,     720,    147,     18,      1;
  7: 128, 1344,  6496,  19152,  38472,  55692,   59906,   48639,   29953,   13923,   4809,   1197,    203,    21,     1;
  8: 256, 3072, 17152,  59136, 140896, 245952,  325360,  332688,  265729,  166344,  81340,  30744,   8806,  1848,   268,   24,   1;
  9: 512, 6912, 43776, 172800, 476352, 973728, 1529472, 1887408, 1854882, 1462563, 927441, 471852, 191184, 60858, 14886, 2700, 342, 27, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+3*x+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[(2 + 3*x + 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] = (2 + 3*x + 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 of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - 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 ?

A200536: Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.

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.