A038324 → 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

\(P_{n}(x) = \left(21 + 30\,x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	21	30														51
2	441	1260	900													2601
3	9261	39690	56700	27000												132651
4	194481	1111320	2381400	2268000	810000											6765201
5	4084101	29172150	83349000	119070000	85050000	24300000										345025251
6	85766121	735138180	2625493500	5000940000	5358150000	3061800000	729000000									17596287801
7	1801088541	18010885410	77189508900	183784545000	262549350000	225042300000	107163000000	21870000000								897410677851
8	37822859361	432261249840	2161306249200	6175160712000	11027072700000	12602368800000	9001692000000	3674160000000	656100000000							45767944570401
9	794280046581	10212172027470	58355268728400	194517562428000	416823348060000	595461925800000	567106596000000	347208120000000	124002900000000	19683000000000						2334165173090451
10	16679880978201	238284013974300	1531825804120500	5835526872840000	14588817182100000	25009400883600000	29773096290000000	24304568400000000	13020304500000000	4133430000000000	590490000000000					119042423827613001
11	350277500542221	5504360722806330	39316862305759500	168500838453255000	481430967009300000	962861934018600000	1375517048598000000	1403588825100000000	1002563446500000000	477411165000000000	136403190000000000	17714700000000000				6071163615208263051
12	7355827511386641	126099900195199560	990784930105139400	4718023476691140000	15165075460792950000	34663029624669600000	57771716041116000000	70740876785040000000	63161497129500000000	40102537860000000000	17186801940000000000	4464104400000000000	531441000000000000			309629344375621415601
13	154472377739119461	2868772729440789990	24589480538063914200	128802040913668122000	460007288977386150000	1182875885941850100000	2253096925603524000000	3218709893719320000000	3448617743270700000000	2736998208945000000000	1563998976540000000000	609350250600000000000	145083393000000000000	15943230000000000000		15791096563156692195651
14	3243919932521508681	64878398650430173620	602442273182565897900	3442527275328947988000	13524214295935152810000	38640612274100436600000	82801312015929507000000	135185815536211440000000	168982269420264300000000	160935494685966000000000	114953924775690000000000	59716324558800000000000	21327258771000000000000	4687309620000000000000	478296900000000000000	805345924720991301978201

Computed in your browser ?

Technical details

Node id: n-b360af258b081e7b · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-0fbe6fa5f58efdcf

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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): 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;
  1:           21,             30;
  2:          441,           1260,            900;
  3:         9261,          39690,          56700,           27000;
  4:       194481,        1111320,        2381400,         2268000,          810000;
  5:      4084101,       29172150,       83349000,       119070000,        85050000,        24300000;
  6:     85766121,      735138180,     2625493500,      5000940000,      5358150000,      3061800000,       729000000;
  7:   1801088541,    18010885410,    77189508900,    183784545000,    262549350000,    225042300000,    107163000000,     21870000000;
  8:  37822859361,   432261249840,  2161306249200,   6175160712000,  11027072700000,  12602368800000,   9001692000000,   3674160000000,    656100000000;
  9: 794280046581, 10212172027470, 58355268728400, 194517562428000, 416823348060000, 595461925800000, 567106596000000, 347208120000000, 124002900000000, 19683000000000;

%F Formula

The row polynomials satisfy P(n,x) = (21+30*x)*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[(21 + 30*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (21 + 30*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-90d6f2d342e10179
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-90d6f2d342e10179
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-90d6f2d342e10179

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-90d6f2d342e10179

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-0ddfaa48a1d96c8a root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - basis_map A013609 e-0f31cc897fc45a50
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*6^(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-0a7984b7b9125974 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
    
    parent_provers periodic_product e-7949976749bfedcf root: theorem check
    
    Parent-row provers of polytool polytool ?
  - parent_provers periodic_product e-96e2fb6c9c1a89f9 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