A200536 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + 9\,x + 2\,x^{2}\right)\,P_{n-1}(x) + \left(9\,x + 4\,x^{2}\right)\,P_{n-1}'(x) + \left(2\,x^{2}\right)\,P_{n-1}''(x)\)

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

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	6	9	2																											17
2	36	189	185	52	4																									466
3	216	3159	7709	5863	1726	204	8																							18885
4	1296	49329	257351	401206	252457	73440	10328	672	16																					1046095
5	7776	751599	7849697	22527895	25602190	13796877	3874034	590520	48560	2000	32																			75051180
6	46656	11343969	229577855	1148703322	2178529729	1940626347	916197893	245786436	38823708	3634000	195952	5568	64																	6713471499
7	279936	170579439	6575744849	55617160963	168364887586	233357647446	170578970681	71721107871	18302998438	2922798900	295260168	18679248	711200	14784	128															727926841637
8	1679616	2561211009	186338948351	2614901064646	12293194450597	25555902429162	27573578032826	17069150831628	6472408635809	1569031697600	249623753744	26384101632	1849354080	84327936	2387712	37888	256													93615012944492
9	10077696	38433281679	5250789656177	120843432453055	866636433993670	2638146784447959	4074536923286012	3572619957180666	1916463377639598	661953377877025	152447399696962	23944460359056	2596660736224	194864037984	10027017408	345112320	7550976	94464	512											14035683284499443
10	60466176	576589924449	147521763190175	5527294751105842	59746218304408729	261978955099512129	567183960411299591	686346865485976212	504464054013780286	238631468687177625	75696311076106833	16575778957598388	2555488956716564	280664439205920	22053428423840	1235092219008	48623899776	1307854080	22775040	230400	1024									2419158497071872087
11	362797056	8649393062319	4138105159275089	251237286953844763	4059618641390579146	25344794961168033648	75778957383850620119	124252099023126271197	122377372206421973366	77179589212274975525	32639526055956039605	9573510341323066167	1995883732543789422	300903459502616124	33170009203218888	2687688460824928	159961276818112	6939150539904	215717094912	4658860800	66102784	551936	2048							473793663877960957858
12	2176782336	129744161108289	115979387295106751	11376042999416815486	273219089568127513837	2408369282074963230144	9836593177218728548364	21561966334127726024364	28000586297843548214067	23094461566545072796675	12726345450409084776337	4865086816096392395970	1327119151103586987577	263805398321280143328	38799203059543730856	4265650743737244576	352566126598768368	21930772958924544	1022496479240192	35362664916480	889970591488	15772123136	185845760	1302528	4096					104412485124909473135549
13	13060694016	1946182007665359	3249109522710961457	513893844046095728215	18271588773258759251950	226014833994023675420001	1250780328625557614505530	3628405162533910050283992	6137577337307437480670926	6525968592751752744857400	4607723287403122770547521	2250933065434462828324113	784953331271082847843190	200131313514854105431509	37990772317338953222738	5442589062749496999432	594044967194181409296	49685962119308534448	3191349024842076384	157131015229393920	5892731655928832	166256859921152	3456382651904	51166881792	508678144	3035136	8192			25675356433361165340926549
14	78364164096	29192847661226529	91000367028127958495	23180461736324409445162	1216723127979091542844129	21025167400475715635975271	156684612798440117846347529	596873770380319686322973268	1302892245759418908416994704	1766396226080327640870135350	1577675368671705633151040648	970383624534456940451880054	425227258263302091926998851	136301327392282183846416471	32621339856348386925486477	5922955291859207987022340	825768519296727487262764	89179692100548136594416	7503161256951939550608	493160999363333421120	25313358490691311040	1010790997854846208	31146150915872512	730462267272192	12755127583744	160088248320	1360801792	6995968	16384	6994166861937904492696686442

Computed in your browser ?

Technical details

Node id: n-e529baa18a6a4626 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-f5044f2210a14593

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:        6,           9,             2;
  2:       36,         189,           185,              52,               4;
  3:      216,        3159,          7709,            5863,            1726,              204,                8;
  4:     1296,       49329,        257351,          401206,          252457,            73440,            10328,              672,               16;
  5:     7776,      751599,       7849697,        22527895,        25602190,         13796877,          3874034,           590520,            48560,            2000,              32;
  6:    46656,    11343969,     229577855,      1148703322,      2178529729,       1940626347,        916197893,        245786436,         38823708,         3634000,          195952,           5568,            64;
  7:   279936,   170579439,    6575744849,     55617160963,    168364887586,     233357647446,     170578970681,      71721107871,      18302998438,      2922798900,       295260168,       18679248,        711200,        14784,         128;
  8:  1679616,  2561211009,  186338948351,   2614901064646,  12293194450597,   25555902429162,   27573578032826,   17069150831628,    6472408635809,   1569031697600,    249623753744,    26384101632,    1849354080,     84327936,     2387712,     37888,     256;
  9: 10077696, 38433281679, 5250789656177, 120843432453055, 866636433993670, 2638146784447959, 4074536923286012, 3572619957180666, 1916463377639598, 661953377877025, 152447399696962, 23944460359056, 2596660736224, 194864037984, 10027017408, 345112320, 7550976, 94464, 512;

%F Formula

The row polynomials satisfy P(n,x) = (6+9*x+2*x^2)*P(n-1,x) + (9*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[(6 + 9*x + 2*x^2)*P[n - 1, x] + (9*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] = (6 + 9*x + 2*x^2)*v[i - 1] + (9*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-4c5ee4490c6f16fe
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-4c5ee4490c6f16fe
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-4c5ee4490c6f16fe

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-4c5ee4490c6f16fe

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-982ad459e6d62a8e root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

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