A038331 → basis A013611

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(17 + 20\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 289 + 680\,x + 400\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	289	680	400															1369
1	4913	17340	20400	8000														50653
2	83521	393040	693600	544000	160000													1874161
3	1419857	8352100	19652000	23120000	13600000	3200000												69343957
4	24137569	170382840	501126000	786080000	693600000	326400000	64000000											2565726409
5	410338673	3379259660	11926798800	23385880000	27512800000	19420800000	7616000000	1280000000										94931877133
6	6975757441	65654187680	270340772800	636095936000	935435200000	880409600000	517888000000	174080000000	25600000000									3512479453921
7	118587876497	1255636339380	5908876891200	16220446368000	28624317120000	33675667200000	26412288000000	13317120000000	3916800000000	512000000000								129961739795077
8	2015993900449	23717575299400	125563633938000	393925126080000	811022318400000	1144972684800000	1122522240000000	754636800000000	332928000000000	87040000000000	10240000000000							4808584372417849
9	34271896307633	443518658098780	2608933282934000	9207999822120000	21665881934400000	35684982009600000	41982331776000000	35279270400000000	20752512000000000	8138240000000000	1914880000000000	204800000000000						177917621779460413
10	582622237229761	8225255113831920	53222238971853600	208714662634720000	552479989327200000	1039962332851200000	1427399280384000000	1439394232320000000	1058378112000000000	553400320000000000	195317760000000000	41779200000000000	4096000000000000					6582952005840035281
11	9904578032905937	151481781679737860	1069283164798149600	4612594044227312000	13566453071256800000	28728959445014400000	45065034423552000000	53017687557120000000	46780312550400000000	30575367680000000000	14388408320000000000	4616601600000000000	905216000000000000	81920000000000000				243569224216081305397
12	168377826559400929	2773281849213662360	21207449435163300400	99799762047827296000	322881583095911840000	759721371990380800000	1340684774100672000000	1802601376942080000000	1855619064499200000000	1455387501568000000000	856110295040000000000	366250393600000000000	107720704000000000000	19496960000000000000	1638400000000000000			9012061295995008299689
13	2862423051509815793	50513347967820278700	415992277382049354000	2120744943516330040000	7484982153587047200000	19372894985754710400000	37986068599519040000000	57457918890028800000000	67597551635328000000000	61853968816640000000000	43661625047040000000000	23348462592000000000000	9156259840000000000000	2485862400000000000000	417792000000000000000	32768000000000000000		333446267951815307088493
14	48661191875666868481	915975376483141053760	8082135674851244592000	44372509587418597760000	169659595481306403200000	479038857829571020800000	1033221065906917888000000	1736505993120870400000000	2298316755601152000000000	2403468502589440000000000	1979327002132480000000000	1270156365004800000000000	622625669120000000000000	225384857600000000000000	56819712000000000000000	8912896000000000000000	655360000000000000000	12337511914217166362274241

Computed in your browser ?

Technical details

Node id: n-21ab4a5c6132df39 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-8c1ae502a0a48e51

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

OEIS matches ?

A009981: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:            289,             680,              400;
  1:           4913,           17340,            20400,             8000;
  2:          83521,          393040,           693600,           544000,            160000;
  3:        1419857,         8352100,         19652000,         23120000,          13600000,           3200000;
  4:       24137569,       170382840,        501126000,        786080000,         693600000,         326400000,          64000000;
  5:      410338673,      3379259660,      11926798800,      23385880000,       27512800000,       19420800000,        7616000000,        1280000000;
  6:     6975757441,     65654187680,     270340772800,     636095936000,      935435200000,      880409600000,      517888000000,      174080000000,       25600000000;
  7:   118587876497,   1255636339380,    5908876891200,   16220446368000,    28624317120000,    33675667200000,    26412288000000,    13317120000000,     3916800000000,     512000000000;
  8:  2015993900449,  23717575299400,  125563633938000,  393925126080000,   811022318400000,  1144972684800000,  1122522240000000,   754636800000000,   332928000000000,   87040000000000,   10240000000000;
  9: 34271896307633, 443518658098780, 2608933282934000, 9207999822120000, 21665881934400000, 35684982009600000, 41982331776000000, 35279270400000000, 20752512000000000, 8138240000000000, 1914880000000000, 204800000000000;

%F Formula

The row polynomials satisfy P(n,x) = (17+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 680*x + 400*x^2.

%t Mathematica

P[0, x_] := 289 + 680*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(17 + 20*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] = 289 + 680*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 20*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)

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-df900b884626232f
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-df900b884626232f
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-df900b884626232f

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 A013611 e-df900b884626232f

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-49423b91d6bf9513 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-b8d4501ecbbc70e3 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[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