Veronese parent of A098172 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(P_{n}(x) = \left(2 + 5\,x + 4\,x^{2} + x^{3}\right)\,P_{n-1}(x)\)

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

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	x²⁹	x³⁰	x³¹	x³²	x³³	x³⁴	x³⁵	x³⁶	x³⁷	x³⁸	x³⁹	x⁴⁰	x⁴¹	x⁴²	x⁴³	P_n(1)
1	1	1																																											2
2	2	7	9	5	1																																								24
3	4	24	61	85	70	34	9	1																																					288
4	8	68	258	575	833	819	553	253	75	13	1																																		3456
5	16	176	888	2720	5641	8361	9108	7380	4446	1966	620	132	17	1																															41472
6	32	432	2720	10600	28610	56695	85305	99385	90585	64790	36234	15674	5140	1235	205	21	1																												497664
7	64	1024	7728	36560	121532	301560	579125	880685	1076010	1065350	858143	562263	298376	127100	42819	11147	2162	294	25	1																									5971968
8	128	2368	20832	115920	457800	1364748	3188738	5984767	9173505	11612615	12227761	10752651	7905989	4853275	2476905	1043165	358435	98805	21315	3465	399	29	1																						71663616
9	256	5376	54016	345600	1580896	5503008	15148336	33830016	62390545	96220561	125197384	138268072	130098892	104474860	71596792	41789944	20693590	8639350	3013560	867160	202188	37228	5208	520	33	1																			859963392
10	512	12032	135936	983040	5111232	20346912	64480896	166994640	360027522	654862247	1014889769	1349795853	1548548241	1537713852	1324231524	989562180	641198928	359503218	173788174	72053110	25433766	7567596	1872468	378180	60726	7455	657	37	1																10319560704
11	1024	26624	334080	2694400	15693440	70318080	252124320	742892640	1833298740	3842321560	6911195501	10753517061	14560497070	17232242090	17881021125	16299685629	13064848704	9207481320	5699450330	3092258890	1465788996	604304636	215371090	65822850	17069820	3703728	659673	93905	10270	810	41	1													123834728448
12	2048	58368	805376	7166720	46221824	230215040	921307200	3043372640	8459876040	20074831700	41110086402	73265596607	114375683009	157192233275	190918758041	205493942313	196384452143	166818969801	125995387705	84576543414	50398155082	26606040142	12407680234	5091508690	1829042886	571419046	153940116	35370907	6832485	1088263	139117	13717	979	45	1										1486016741376
13	4096	126976	1910784	18595840	131557120	721011456	3185743616	11660363264	36052059120	95543841360	219477208104	440840828064	779594526361	1220435354425	1698567011100	2108726290940	2341105881290	2328454727610	2077117375300	1662790356700	1194479547855	769504417055	444014724920	229043734200	105352390300	43061767516	15572655696	4955161464	1377699015	331762695	68420396	11908556	1715274	198970	17860	1164	49	1							17832200896512
14	8192	274432	4472832	47257600	363863552	2176102656	10521368832	42265047552	143869920480	421175174992	1072542222752	2397295121208	4736845866818	8321683861015	13058529727833	18371623581441	23240546616105	26495911036530	27269658204750	25366092381630	21339835108020	16239685375485	11176260083235	6950658309075	3901486768335	1973075148252	896907444372	365373061772	132883595650	42945322017	12261611791	3070668887	668417607	125028930	19800222	2602782	276328	22763	1365	53	1				213986410758144
15	16384	589824	10350592	117985280	982180864	6365026304	33425962752	146205213440	543326656704	1741281511424	4868705049936	11985871976624	26171511405676	50989319763704	89070157349221	140059477112925	198895013911762	255739579207182	298352681638011	316304666549235	305084675979720	267912574022340	214276379857605	156081193644285	103500990790470	62432477457714	34215796012419	17009070706747	7653337425900	3108708313744	1136157278039	372114280447	108681948830	28136433234	6409084409	1272740001	217796384	31638516	3824639	374311	28490	1582	57	1	2567836929097728

Technical details

Node id: n-a9043bb05acfd8df · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-9835034ff7571c00

Observed on the stored rows ?

Degrees: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43 (affine: slope 3, intercept -2)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   1,     1;
   2:   2,     7,      9,      5,       1;
   3:   4,    24,     61,     85,      70,       34,        9,         1;
   4:   8,    68,    258,    575,     833,      819,      553,       253,        75,        13,          1;
   5:  16,   176,    888,   2720,    5641,     8361,     9108,      7380,      4446,      1966,        620,        132,         17,          1;
   6:  32,   432,   2720,  10600,   28610,    56695,    85305,     99385,     90585,     64790,      36234,      15674,       5140,       1235,        205,        21,         1;
   7:  64,  1024,   7728,  36560,  121532,   301560,   579125,    880685,   1076010,   1065350,     858143,     562263,     298376,     127100,      42819,     11147,      2162,       294,        25,        1;
   8: 128,  2368,  20832, 115920,  457800,  1364748,  3188738,   5984767,   9173505,  11612615,   12227761,   10752651,    7905989,    4853275,    2476905,   1043165,    358435,     98805,     21315,     3465,      399,      29,       1;
   9: 256,  5376,  54016, 345600, 1580896,  5503008, 15148336,  33830016,  62390545,  96220561,  125197384,  138268072,  130098892,  104474860,   71596792,  41789944,  20693590,   8639350,   3013560,   867160,   202188,   37228,    5208,    520,    33,    1;
  10: 512, 12032, 135936, 983040, 5111232, 20346912, 64480896, 166994640, 360027522, 654862247, 1014889769, 1349795853, 1548548241, 1537713852, 1324231524, 989562180, 641198928, 359503218, 173788174, 72053110, 25433766, 7567596, 1872468, 378180, 60726, 7455, 657, 37, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+5*x+4*x^2+x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1 + x.

%t Mathematica

P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(2 + 5*x + 4*x^2 + x^3)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + x; for(n = 2, N, my(i = n); v[i] = (2 + 5*x + 4*x^2 + x^3)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
  - 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
  - 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
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?

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.