A038324 → 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

\(P_{n}(x) = \left(21 + 40\,x\right)\,P_{n-1}(x)\)

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

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	40														61
2	441	1680	1600													3721
3	9261	52920	100800	64000												226981
4	194481	1481760	4233600	5376000	2560000											13845841
5	4084101	38896200	148176000	282240000	268800000	102400000										844596301
6	85766121	980184240	4667544000	11854080000	16934400000	12902400000	4096000000									51520374361
7	1801088541	24014513880	137225793600	435637440000	829785600000	948326400000	602112000000	163840000000								3142742836021
8	37822859361	576348333120	3842322220800	14637417984000	34850995200000	53106278400000	50577408000000	27525120000000	6553600000000							191707312997281
9	794280046581	13616229369960	103742699961600	461078666496000	1317367618560000	2509271654400000	3186376704000000	2601123840000000	1238630400000000	262144000000000						11694146092834141
10	16679880978201	317712018632400	2723245873992000	13832359994880000	46107866649600000	105389409484800000	167284776960000000	182078668800000000	130056192000000000	55050240000000000	10485760000000000					713342911662882601
11	350277500542221	7339147630408440	69896644099128000	399409394852160000	1521559599436800000	4057492265164800000	7728556695552000000	10515043123200000000	10014326784000000000	6358302720000000000	2422210560000000000	419430400000000000				43513917611435838661
12	7355827511386641	168133200260266080	1761395431298025600	11183463055860480000	47929127382259200000	146069721545932800000	324599381213184000000	529958173409280000000	630902587392000000000	534097428480000000000	305198530560000000000	105696460800000000000	16777216000000000000			2654348974297586158321
13	154472377739119461	3825030305921053320	43714632067669180800	305308541424991104000	1453850197261862400000	4984629247754956800000	12659375867314176000000	24113096890122240000000	34447281271603200000000	36452149493760000000000	27773066280960000000000	14427566899200000000000	4580179968000000000000	671088640000000000000		161915287432152755657581
14	3243919932521508681	86504531533906898160	1071008485657894929600	8160064652631580416000	42743195799498754560000	162831222093328588800000	465232063123795968000000	1012750069385134080000000	1687916782308556800000000	2143386390233088000000000	2041320371650560000000000	1413901556121600000000000	673286455296000000000000	197300060160000000000000	26843545600000000000000	9876832533361318095112441

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Technical details

Node id: n-e9f99a7290620624 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-ae9bc19c6d2e3d8d

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,             40;
  2:          441,           1680,            1600;
  3:         9261,          52920,          100800,           64000;
  4:       194481,        1481760,         4233600,         5376000,          2560000;
  5:      4084101,       38896200,       148176000,       282240000,        268800000,        102400000;
  6:     85766121,      980184240,      4667544000,     11854080000,      16934400000,      12902400000,       4096000000;
  7:   1801088541,    24014513880,    137225793600,    435637440000,     829785600000,     948326400000,     602112000000,     163840000000;
  8:  37822859361,   576348333120,   3842322220800,  14637417984000,   34850995200000,   53106278400000,   50577408000000,   27525120000000,    6553600000000;
  9: 794280046581, 13616229369960, 103742699961600, 461078666496000, 1317367618560000, 2509271654400000, 3186376704000000, 2601123840000000, 1238630400000000, 262144000000000;

%F Formula

The row polynomials satisfy P(n,x) = (21+40*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 + 40*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 + 40*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-bf7e65720bb019eb
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-bf7e65720bb019eb
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-bf7e65720bb019eb

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-bf7e65720bb019eb

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