A139548 → basis A008277

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(P_{n}(x) = \left(8 + 10\,x + 2\,x^{2}\right)\,P_{n-1}(x) + \left(10\,x + 4\,x^{2}\right)\,P_{n-1}'(x) + \left(2\,x^{2}\right)\,P_{n-1}''(x)\)

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

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	8	10	2																											20
2	64	260	220	56	4																									604
3	512	5320	10808	7280	1960	216	8																							26104
4	4096	100880	421360	569184	322896	86688	11424	704	16																					1517248
5	32768	1856800	14904032	36245440	36613280	18029088	4708704	675840	52800	2080	32																			113120864
6	262144	33750080	502989760	2084258176	3463363904	2791286784	1216813312	305640192	45659328	4072640	210496	5760	64																	10448312640
7	2097152	610122880	16568697728	113334224640	296221860480	367762082688	246417435264	96467937280	23179819520	3514551040	339282944	20616960	757120	15232	128															1164439501056
8	16777216	11003183360	538744241920	5966168036864	23854464292096	43969120925184	43166737915392	24741522803712	8794580391168	2017009646080	305742474752	30961781760	2088672768	92001280	2524160	38912	256													153398255706880
9	134217728	198225072640	17393893862912	308027804564480	1849854614325760	4941122258336256	6892125009509888	5563734025175040	2785427420605440	906971756387840	198425437230592	29786225264640	3102180889600	224467160064	11173794816	373088256	7938048	96768	512											23496405007521280
10	1073741824	3569393484800	559380023065600	15714876767901696	140000931719599104	532942256898551808	1034125378196004864	1145539406918365184	782291627096759296	347602584761635840	104438081557462016	21802959513753600	3221361947474944	340509164154880	25842308648960	1402049789952	53607622656	1403458560	23833600	235520	1024									4128610224425545728
11	8589934592	64259820144640	17950134394370048	795819817597235200	10426913132751104000	55899795154303457280	148636317979484375040	221803531293236428800	202057073638641745920	119267152664216494080	47632469641777827840	13286509563045089280	2649009860267796480	383674035850749952	40785554100819968	3196995477733376	184565842075648	7784360736768	235751892992	4969000960	68913152	563200	2048							822900456324548491264
12	68719476736	1156762661949440	575303955281735680	40114221818600710144	768281681814007402496	5750871306208461987840	20723296382268605931520	41097606471115377131520	49137221552830302228480	37792655100149817753600	19608554426663256637440	7110998019726564556800	1851148320912966696960	352868907154459852800	49968013642901913600	5307058028485509120	424959457307996160	25672171721011200	1164913336012800	39283035545600	965545799680	16736051200	193126400	1327104	4096					184289919759928348348416
13	549755813888	20822415109857280	18425921383721787392	2016068875650430648320	56199944578528312074240	583640940179230668693504	2826644639480263433674752	7373906067930240620953600	11430599949232981988556800	11291781052397359344230400	7484000252591570132705280	3459530362201041287577600	1148861143716443944550400	280379606871912131297280	51163791870568224030720	7071106255205070274560	746803817615370977280	60596423880332820480	3784278977665105920	181514874346291200	6642816063037440	183171935232000	3726762024960	54057369600	527155200	3088384	8192			45996624745790461239836672
14	4398046511104	374808969535569920	589920999190146826240	101135152012258744303616	4090786376761115387183104	58662042834357642507190272	379432141948007696195321856	1291845982627386398676811776	2571981709706906068174946304	3227444024683806686065049600	2697675201055116692030832640	1566104837158325180524953600	652136208400885396058357760	199723830152316290442854400	45877119351222370733260800	8024668892709857667317760	1081233895420251760558080	113156313055547889254400	9247551880028786688000	591599843102633164800	29609083098131496960	1154679101841408000	34797023890636800	799138260910080	13680089579520	168496054272	1406877696	7110656	16384	12704304538795197354131046400

Computed in your browser ?

Technical details

Node id: n-f2ca92e21d86d85e · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-910fa4df8c34a164

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): no

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         1;
  1:         8,           10,              2;
  2:        64,          260,            220,              56,                4;
  3:       512,         5320,          10808,            7280,             1960,              216,                8;
  4:      4096,       100880,         421360,          569184,           322896,            86688,            11424,              704,               16;
  5:     32768,      1856800,       14904032,        36245440,         36613280,         18029088,          4708704,           675840,            52800,            2080,              32;
  6:    262144,     33750080,      502989760,      2084258176,       3463363904,       2791286784,       1216813312,        305640192,         45659328,         4072640,          210496,           5760,            64;
  7:   2097152,    610122880,    16568697728,    113334224640,     296221860480,     367762082688,     246417435264,      96467937280,      23179819520,      3514551040,       339282944,       20616960,        757120,        15232,         128;
  8:  16777216,  11003183360,   538744241920,   5966168036864,   23854464292096,   43969120925184,   43166737915392,   24741522803712,    8794580391168,   2017009646080,    305742474752,    30961781760,    2088672768,     92001280,     2524160,     38912,     256;
  9: 134217728, 198225072640, 17393893862912, 308027804564480, 1849854614325760, 4941122258336256, 6892125009509888, 5563734025175040, 2785427420605440, 906971756387840, 198425437230592, 29786225264640, 3102180889600, 224467160064, 11173794816, 373088256, 7938048, 96768, 512;

%F Formula

The row polynomials satisfy P(n,x) = (8+10*x+2*x^2)*P(n-1,x) + (10*x+4*x^2)*d/dx P(n-1,x) + 2*x^2*d^2/dx^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[(8 + 10*x + 2*x^2)*P[n - 1, x] + (10*x + 4*x^2)*D[P[n - 1, x], x] + 2*x^2*D[P[n - 1, x], {x, 2}]];
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] = (8 + 10*x + 2*x^2)*v[i - 1] + (10*x + 4*x^2)*deriv(v[i - 1], x) + 2*x^2*deriv(deriv(v[i - 1], x), x)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-b4af3e745574d56e
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-b4af3e745574d56e
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-b4af3e745574d56e

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A008277 e-b4af3e745574d56e

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-4755dd2fd7289161 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A034870 → basis A008277

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741