A038321 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(39 + 63\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1521 + 4914\,x + 3969\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1521	4914	3969															10404
1	59319	287469	464373	250047														1061208
2	2313441	14948388	36221094	39007332	15752961													108243216
3	90224199	728733915	2354371110	3803214870	3071827395	992436543												11040808032
4	3518743761	34104747222	137730709935	296650759860	359403805215	232230151062	62523502209											1126162419264
5	137231006679	1551765998601	7520096762451	20246414360445	32705746274565	31699415619963	17068916103057	3938980639167										114868566764928
6	5352009260481	69164427366216	391045031647452	1263376256091768	2551048209416070	3296739224476152	2662750912076892	1228961959420104	248155780267521									11716593810022656
7	208728361158759	3034589250692727	19608115158322236	73907510981368428	179083584301008114	289288866947782338	311541856712996364	215682823878228252	87102678873899871	15633814156853823								1195092568622310912
8	8140406085191601	131498867530018170	955895613968209005	4117704183247669560	11640432979565527410	22564531621927022364	30375331029517145490	28038767104169672760	16985022380410474845	6097187521172990970	984930291881790849							121899441999475713024
9	317475837322472439	5641301417037779493	45564357599151295905	220811886826656280155	713392249747658751270	1613364010967782099026	2606203402332571083042	3007157771922197403510	2428858200398697902835	1307846723291606563065	422535095217288274221	62050608388552823487						12433743083946522728448
10	12381557655576425121	240011733015789163884	2132411935640280648354	11482218114986126568060	41733446610238036949295	107864908161846003192024	203283865381940544477276	281469967451917676968536	284176409446647654631695	204024088833490623838140	98873212280845456167714	29039684725842721391916	3909188328478827879681					1268241794562545318301696
11	482880748567480579719	10140495719917092174099	98284804669965662610498	582148458429796617000642	2350984159043409414810285	6835938554756990452294521	14723559964091979435711276	23784212249687043703841292	28815487917890072179653873	25860053259644936571484245	16709572875462882092343666	7361560078001129872850706	1981958482538765734998267	246278864694166156419903				129360663045379622466772992
12	18832349194131742609041	425900820236517871312158	4471958612483437648777659	28895732572969904807486412	128363735083770154048641561	414713605655257420772534274	1004882967549277596487294587	1855168555475589408899620776	2622209400527996568348502443	2823917815953227073606079554	2280856697500683405604910409	1339803934196205636858828492	541074665733083045654526891	134468260123014721405267038	15515568475732467854453889			13194787630628721491610845184
13	734461618571137961752599	17796569988454496765543745	201238137561754694194994655	1408666962932282859364962585	6826616820364140010768664835	24260745930832559115193255029	65317392890703043771674148155	135659200619152475525784769245	219141785615553998926267704165	275331987055439639676592756515	266860233607579958455774517853	195946325376195074390603666955	105509559817951193902632743745	39331966085981806011040608615	9076607558303493694855525065	977480813971145474830595007		1345868338324129592144306208768
14	28644003124274380508351361	740337311519707065446619792	8969471274181066369834047480	67616014220749577249518204080	354984074658935280559970571420	1376245950985410626170962830736	4075805316379869931352466844872	9405704576261238303121077334320	17093059278013211916248880924870	24543879988942047879741982866480	27753464295188315679400549856712	24454101406949145283947337635984	16459491331600386248810708024220	8181048945884215650296446591920	2831901558190690032794923820280	609948027917994776294291284368	61581291280182164914327485441	137278570509061218398719233294336

Technical details

Node id: n-20046f09233a3fbb · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-8f86ccbca4a04f09

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:               1521,                4914,                 3969;
  1:              59319,              287469,               464373,                250047;
  2:            2313441,            14948388,             36221094,              39007332,              15752961;
  3:           90224199,           728733915,           2354371110,            3803214870,            3071827395,              992436543;
  4:         3518743761,         34104747222,         137730709935,          296650759860,          359403805215,           232230151062,            62523502209;
  5:       137231006679,       1551765998601,        7520096762451,        20246414360445,        32705746274565,         31699415619963,         17068916103057,          3938980639167;
  6:      5352009260481,      69164427366216,      391045031647452,      1263376256091768,      2551048209416070,       3296739224476152,       2662750912076892,       1228961959420104,        248155780267521;
  7:    208728361158759,    3034589250692727,    19608115158322236,     73907510981368428,    179083584301008114,     289288866947782338,     311541856712996364,     215682823878228252,      87102678873899871,      15633814156853823;
  8:   8140406085191601,  131498867530018170,   955895613968209005,   4117704183247669560,  11640432979565527410,   22564531621927022364,   30375331029517145490,   28038767104169672760,   16985022380410474845,    6097187521172990970,    984930291881790849;
  9: 317475837322472439, 5641301417037779493, 45564357599151295905, 220811886826656280155, 713392249747658751270, 1613364010967782099026, 2606203402332571083042, 3007157771922197403510, 2428858200398697902835, 1307846723291606563065, 422535095217288274221, 62050608388552823487;

%F Formula

The row polynomials satisfy P(n,x) = (39+63*x)*P(n-1,x) for n >= 1, with P(0,x) = 1521 + 4914*x + 3969*x^2.

%t Mathematica

P[0, x_] := 1521 + 4914*x + 3969*x^2;
P[n_, x_] := P[n, x] = Expand[(39 + 63*x)*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] = 1521 + 4914*x + 3969*x^2; for(n = 1, N, my(i = n + 1); v[i] = (39 + 63*x)*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 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)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    
    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
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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

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.