Veronese parent of A098172 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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	x⁴⁴	P_n(1)
1	1	2	1																																											4
2	2	9	16	14	6	1																																								48
3	4	28	85	146	155	104	43	10	1																																					576
4	8	76	326	833	1408	1652	1372	806	328	88	14	1																																		6912
5	16	192	1064	3608	8361	14002	17469	16488	11826	6412	2586	752	149	18	1																															82944
6	32	464	3152	13320	39210	85305	142000	184690	189970	155375	101024	51908	20814	6375	1440	226	22	1																												995328
7	64	1088	8752	44288	158092	423092	880685	1459810	1956695	2141360	1923493	1420406	860639	425476	169919	53966	13309	2456	319	26	1																									11943936
8	128	2496	23200	136752	573720	1822548	4553486	9173505	15158272	20786120	23840376	22980412	18658640	12759264	7330180	3520070	1401600	457240	120120	24780	3864	428	30	1																						143327232
9	256	5632	59392	399616	1926496	7083904	20651344	48978352	96220561	158611106	221417945	263465456	268366964	234573752	176071652	113386736	62483534	29332940	11652910	3880720	1069348	239416	42436	5728	553	34	1																			1719926784
10	512	12544	147968	1118976	6094272	25458144	84827808	231475536	527022162	1014889769	1669752016	2364685622	2898344094	3086262093	2861945376	2313793704	1630761108	1000702146	533291392	245841284	97486876	33001362	9440064	2250648	438906	68181	8112	694	38	1																20639121408
11	1024	27648	360704	3028480	18387840	86011520	322442400	995016960	2576191380	5675620300	10753517061	17664712562	25314014131	31792739160	35113263215	34180706754	29364534333	22272330024	14906931650	8791709220	4558047886	2070093632	819675726	281193940	82892670	20773548	4363401	753578	104175	11080	851	42	1													247669456896
12	2048	60416	863744	7972096	53388544	276436864	1151522240	3964679840	11503248680	28534707740	61184918102	114375683009	187641279616	271567916284	348110991316	396412700354	401878394456	363203421944	292814357506	210571931119	134974698496	77004195224	39013720376	17499188924	6920551576	2400461932	725359162	189311023	42203392	7920748	1227380	152834	14696	1024	46	1										2972033482752
13	4096	131072	2037760	20506624	150152960	852568576	3906755072	14846106880	47712422384	131595900480	315021049464	660318036168	1220435354425	2000029880786	2919002365525	3807293302040	4449832172230	4669560608900	4405572102910	3739907732000	2857269904555	1963983964910	1213519141975	673058459120	334396124500	148414157816	58634423212	20527817160	6332860479	1709461710	400183091	80328952	13623830	1914244	216830	19024	1213	50	1							35664401793024
14	8192	282624	4747264	51730432	411121152	2539966208	12697471488	52786416384	186134968032	565045095472	1493717397744	3469837343960	7134140988026	13058529727833	21380213588848	31430153309274	41612170197546	49736457652635	53765569241280	52635750586380	46705927489650	37579520483505	27415945458720	18126918392310	10852145077410	5874561916587	2869982592624	1262280506144	498256657422	175828917667	55206933808	15332280678	3739086494	793446537	144829152	22403004	2879110	299091	24128	1418	54	1				427972821516288
15	16384	606208	10940416	128335872	1100166144	7347207168	39790989056	179631176192	689531870144	2284608168128	6609986561360	16854577026560	38157383382300	77160831169380	140059477112925	229129634462146	338954491024687	454634593118944	554092260845193	614657348187246	621389342528955	572997250002060	482188953879945	370357573501890	259582184434755	165933468248184	96648273470133	51224866719166	24662408132647	10762045739644	4244865591783	1508271558486	480796229277	136818382064	34545517643	7681824410	1490536385	249434900	35463155	4198950	402801	30072	1639	58	1	5135673858195456

Technical details

Node id: n-256337cf537e7b19 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-37798f32d2e092fd

Observed on the stored rows ?

Degrees: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44 (affine: slope 3, intercept -1)
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,     2,      1;
   2:   2,     9,     16,      14,       6,        1;
   3:   4,    28,     85,     146,     155,      104,       43,        10,         1;
   4:   8,    76,    326,     833,    1408,     1652,     1372,       806,       328,         88,         14,          1;
   5:  16,   192,   1064,    3608,    8361,    14002,    17469,     16488,     11826,       6412,       2586,        752,        149,         18,          1;
   6:  32,   464,   3152,   13320,   39210,    85305,   142000,    184690,    189970,     155375,     101024,      51908,      20814,       6375,       1440,        226,         22,          1;
   7:  64,  1088,   8752,   44288,  158092,   423092,   880685,   1459810,   1956695,    2141360,    1923493,    1420406,     860639,     425476,     169919,      53966,      13309,       2456,       319,        26,        1;
   8: 128,  2496,  23200,  136752,  573720,  1822548,  4553486,   9173505,  15158272,   20786120,   23840376,   22980412,   18658640,   12759264,    7330180,    3520070,    1401600,     457240,    120120,     24780,     3864,      428,      30,       1;
   9: 256,  5632,  59392,  399616, 1926496,  7083904, 20651344,  48978352,  96220561,  158611106,  221417945,  263465456,  268366964,  234573752,  176071652,  113386736,   62483534,   29332940,  11652910,   3880720,  1069348,   239416,   42436,    5728,    553,    34,    1;
  10: 512, 12544, 147968, 1118976, 6094272, 25458144, 84827808, 231475536, 527022162, 1014889769, 1669752016, 2364685622, 2898344094, 3086262093, 2861945376, 2313793704, 1630761108, 1000702146, 533291392, 245841284, 97486876, 33001362, 9440064, 2250648, 438906, 68181, 8112, 694, 38, 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 + 2*x + x^2.

%t Mathematica

P[1, x_] := 1 + 2*x + x^2;
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 + 2*x + x^2; 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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (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.