A299989 → basis A013610

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 18\,x + 9\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 3\,x\)

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	P_n(1)
0	1	3																													4
1	8	42	63	27																											140
2	64	480	1332	1728	1053	243																									4900
3	512	4992	19872	42120	51516	36450	13851	2187																							171500
4	4096	49152	253440	739584	1349136	1597968	1230552	594864	164025	19683																					6002500
5	32768	466944	2949120	10920960	26386560	43724448	50750064	41290560	23094720	8463690	1830519	177147																			210087500
6	262144	4325376	32292864	144654336	434211840	923042304	1430519616	1637345664	1384738416	855029520	374843052	110539728	19663317	1594323																	7353062500
7	2097152	39321600	338558976	1777434624	6368108544	16502040576	31966824960	47155499136	53424805824	46501638624	30851921520	15326758440	5520609108	1361551842	205667667	14348907															257357187500
8	16777216	352321536	3435134976	20667432960	85985722368	262639190016	610084306944	1101165207552	1563898855680	1758059106048	1564668119808	1096463402496	597713818464	248204204640	75838756464	16070775840	2109289329	129140163													9007501562500
9	134217728	3120562176	33973862400	230342787072	1090815787008	3834863419392	10382051377152	22154591895552	37822923343872	52125139118592	58237498568448	52758265330944	38600064870912	22612652991936	10453810101408	3727501664832	988697087928	183637311786	21308126895	1162261467											315262554687500
10	1073741824	27380416512	329168977920	2482356879360	13178461224960	52386676604928	161901294649344	398627430727680	794804503265280	1297205060198400	1744558802777088	1939467348946944	1782586782040320	1350526779590400	838058818504320	421502472071424	169088897583072	52813161060480	12374210418660	2045580181920	212693848461	10460353203									11034189414062500
11	8589934592	238370684928	3135862996992	26030320386048	153072634429440	678646926802944	2356776687108096	6574722838953984	14990841431064576	28271768416911360	44459402035175424	58592642783348736	64872135762361344	60345982453971456	47057233619024640	30611819525962752	16482285044489088	7259627693621952	2571430660685568	714418878535560	149889887830188	22322393735202	2102530993803	94143178827							386196629492187500
12	68719476736	2061584302080	29454885715968	266833433198592	1721349609357312	8418755717627904	32447511889182720	101127585420804096	259482732733661184	555181798645039104	999484620665389056	1523456294652149760	1973779274515746816	2177800088404414464	2046534774984935424	1635038603435888640	1106386134394464000	630264528083443968	299585309171081472	117337752163222272	37201534862451696	9306367037645040	1767632325655752	239500246935888	20617356163113	847288609443					13516882032226562500
13	549755813888	17729624997888	273366078455808	2683409667194880	18838892643876864	100735839608242176	426609844514979840	1468844698830372864	4188186006446407680	10022291847153451008	20324493935536766976	35175009716999553024	52207809085853171712	66661534300388106240	73336693261800665088	69518135572855676928	56700596911866126336	39672414074690901504	23698919188420819200	12003618335136240384	5105959600377347712	1800118333294291296	516469479044922000	117490687176096000	20384634225052656	2533392942234570	200807400437991	7625597484987			473090871127929687500
14	4398046511104	151732604633088	2511009679933440	26547433374744576	201472809866625024	1169137471460474880	5395673902863089664	20336357348386799616	63784181231152791552	168785285182736302080	380690878791073923072	737441595200039288832	1233733093012648230912	1789607925401457917952	2256471445274069778432	2476159371998750343168	2364961454962537181184	1963653277166818578432	1414000179058597917696	879661218744882782208	470203079531258481408	214340784489372752640	82487522235052748736	26437441119865985664	6926134754374447248	1444606744173688368	230669240189200092	26476074467874864	1944527358671685	68630377364883	16558180489477539062500

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Technical details

Node id: n-8624395344cd3600 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-05ba62c68bfd52e0

Observed on the stored rows ?

Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, 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

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,          3;
  1:         8,         42,          63,           27;
  2:        64,        480,        1332,         1728,          1053,           243;
  3:       512,       4992,       19872,        42120,         51516,         36450,          13851,           2187;
  4:      4096,      49152,      253440,       739584,       1349136,       1597968,        1230552,         594864,         164025,          19683;
  5:     32768,     466944,     2949120,     10920960,      26386560,      43724448,       50750064,       41290560,       23094720,        8463690,        1830519,         177147;
  6:    262144,    4325376,    32292864,    144654336,     434211840,     923042304,     1430519616,     1637345664,     1384738416,      855029520,      374843052,      110539728,       19663317,        1594323;
  7:   2097152,   39321600,   338558976,   1777434624,    6368108544,   16502040576,    31966824960,    47155499136,    53424805824,    46501638624,    30851921520,    15326758440,     5520609108,     1361551842,      205667667,      14348907;
  8:  16777216,  352321536,  3435134976,  20667432960,   85985722368,  262639190016,   610084306944,  1101165207552,  1563898855680,  1758059106048,  1564668119808,  1096463402496,   597713818464,   248204204640,    75838756464,   16070775840,   2109289329,    129140163;
  9: 134217728, 3120562176, 33973862400, 230342787072, 1090815787008, 3834863419392, 10382051377152, 22154591895552, 37822923343872, 52125139118592, 58237498568448, 52758265330944, 38600064870912, 22612652991936, 10453810101408, 3727501664832, 988697087928, 183637311786, 21308126895, 1162261467;

%F Formula

The row polynomials satisfy P(n,x) = (8+18*x+9*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 3*x.

%t Mathematica

P[0, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[(8 + 18*x + 9*x^2)*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] = 1 + 3*x; for(n = 1, N, my(i = n + 1); v[i] = (8 + 18*x + 9*x^2)*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)

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-78fc61375bb1ec88
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-78fc61375bb1ec88
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-78fc61375bb1ec88

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 A013610 e-78fc61375bb1ec88

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) f…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-0e7ef206328a472c root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.