A038305 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(22 + 27\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 484 + 1188\,x + 729\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	484	1188	729															2401
1	10648	39204	48114	19683														117649
2	234256	1149984	2117016	1732104	531441													5764801
3	5153632	31624560	77623920	95265720	58458510	14348907												282475249
4	113379904	834888384	2561589360	4191691680	3858261660	1894055724	387420489											13841287201
5	2494357888	21428801856	78896952288	161380129680	198057431880	145842290748	59662755306	10460353203										678223072849
6	54875873536	538781303808	2314310600448	5680580564736	8714527002720	8556081057216	5250322466928	1841022163728	282429536481									33232930569601
7	1207269217792	13334837269248	65461928412672	187459158636288	345095269307712	423526012332192	346521282817248	182261194209072	55921048223238	7625597484987								1628413597910449
8	26559922791424	325962688803840	1800203031348480	5891573557140480	12653493207949440	18635144542616448	19058670554948640	13365820908665280	6151315304556180	1677631446697140	205891132094649							79792266297612001
9	584318301411328	7888297069052928	48405459287370240	178220100103499520	437449336617680640	751617496552196736	922439654859514176	808632164974249440	496206101234198520	202993405050353940	49825653966905058	5559060566555523						3909821048582988049
10	12855002631049216	189319129657270272	1277904125186574336	5227789603035985920	14435828108383461120	28346717012825705472	40587344813818623744	42695778310640370432	32749602681457102320	17863419644431146720	6576986323631467656	1467591989570658072	150094635296999121					191581231380566414401
11	282810057883082752	4512105923498274816	33225507254850932736	149514782646829197312	458738537666407764480	1013395133208518970624	1658282945250303770112	2035165432807190990592	1873277273379346252704	1277234504576826990480	627006029519533249872	209865654508604104296	42927065694941748606	4052555153018976267				9387480337647754305649
12	6221821273427820544	106902201879805280256	852788019541173940224	4186413914111217524736	14129146960125359145984	34680633447580426994688	63843893392136695149312	89547279043516403586048	96161566700139774305472	78677645481932542613568	48279464273004060240144	21546207196216688041056	6610768117021029285324	1248186987129844690236	109418989131512359209			459986536544739960976801
13	136880068015412051968	2519837615738267320320	21647695880660569251840	115126382638058481930240	423874408803760774379520	1144460903770154090824704	2340942757711678822141440	3693825260545051647924480	4533331001578017931543680	4327270501506289843746240	3186444642018267975849504	1777562093687876763387120	727184492872313221385640	205950852876424373888940	36108266413399078538970	2954312706550833698643		22539340290692258087863249
14	3011361496339065143296	59132189382658006450176	544284924999465741189120	3117268206815121972264960	12433649324910316048465920	36622748920644930906390528	82401185071451094539378688	144469610190206464452157440	199466564069432788987921920	217599888075744860714096640	186938085665071721249837568	125140371395626524142453248	63992235372763563481936320	24164900070833793202968960	6355054888758237822858720	1039918072705893461922336	79766443076872509863361	1104427674243920646305299201

Technical details

Node id: n-2f306d780dbae501 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-b7d0fa5aa93aa995

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:             484,             1188,               729;
  1:           10648,            39204,             48114,              19683;
  2:          234256,          1149984,           2117016,            1732104,             531441;
  3:         5153632,         31624560,          77623920,           95265720,           58458510,           14348907;
  4:       113379904,        834888384,        2561589360,         4191691680,         3858261660,         1894055724,          387420489;
  5:      2494357888,      21428801856,       78896952288,       161380129680,       198057431880,       145842290748,        59662755306,        10460353203;
  6:     54875873536,     538781303808,     2314310600448,      5680580564736,      8714527002720,      8556081057216,      5250322466928,      1841022163728,       282429536481;
  7:   1207269217792,   13334837269248,    65461928412672,    187459158636288,    345095269307712,    423526012332192,    346521282817248,    182261194209072,     55921048223238,      7625597484987;
  8:  26559922791424,  325962688803840,  1800203031348480,   5891573557140480,  12653493207949440,  18635144542616448,  19058670554948640,  13365820908665280,   6151315304556180,   1677631446697140,   205891132094649;
  9: 584318301411328, 7888297069052928, 48405459287370240, 178220100103499520, 437449336617680640, 751617496552196736, 922439654859514176, 808632164974249440, 496206101234198520, 202993405050353940, 49825653966905058, 5559060566555523;

%F Formula

The row polynomials satisfy P(n,x) = (22+27*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1188*x + 729*x^2.

%t Mathematica

P[0, x_] := 484 + 1188*x + 729*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 27*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] = 484 + 1188*x + 729*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 27*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)*10^(i-j)*3^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool 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.