A038334 → 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

\(P_{n}(x) = \left(20 + 24\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 400 + 960\,x + 576\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	400	960	576								1936
1	8000	28800	34560	13824							85184
2	160000	768000	1382400	1105920	331776						3748096
3	3200000	19200000	46080000	55296000	33177600	7962624					164916224
4	64000000	460800000	1382400000	2211840000	1990656000	955514880	191102976				7256313856
5	1280000000	10752000000	38707200000	77414400000	92897280000	66886041600	26754416640	4586471424			319277809664
6	25600000000	245760000000	1032192000000	2477260800000	3715891200000	3567255552000	2140353331200	733835427840	110075314176		14048223625216
7	512000000000	5529600000000	26542080000000	74317824000000	133772083200000	160526499840000	128421199872000	66045188505600	19813556551680	2641807540224	618121839509504

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-6081e8199e2268e3 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-da2debf48cb6165b

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 ?

A009988: 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:             400,              960,               576;
  1:            8000,            28800,             34560,             13824;
  2:          160000,           768000,           1382400,           1105920,             331776;
  3:         3200000,         19200000,          46080000,          55296000,           33177600,            7962624;
  4:        64000000,        460800000,        1382400000,        2211840000,         1990656000,          955514880,          191102976;
  5:      1280000000,      10752000000,       38707200000,       77414400000,        92897280000,        66886041600,        26754416640,         4586471424;
  6:     25600000000,     245760000000,     1032192000000,     2477260800000,      3715891200000,      3567255552000,      2140353331200,       733835427840,       110075314176;
  7:    512000000000,    5529600000000,    26542080000000,    74317824000000,    133772083200000,    160526499840000,    128421199872000,     66045188505600,     19813556551680,     2641807540224;
  8:  10240000000000,  122880000000000,   663552000000000,  2123366400000000,   4459069440000000,   6421059993600000,   6421059993600000,   4403012567040000,   1981355655168000,   528361508044800,    63403380965376;
  9: 204800000000000, 2703360000000000, 16220160000000000, 58392576000000000, 140142182400000000, 235438866432000000, 282526639718400000, 242165691187200000, 145299414712320000, 58119765884928000, 13948743812382720, 1521681143169024;

%F Formula

The row polynomials satisfy P(n,x) = (20+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 960*x + 576*x^2.

%t Mathematica

P[0, x_] := 400 + 960*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 24*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] = 400 + 960*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 24*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-889e30cc3b49f76a
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-889e30cc3b49f76a
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-889e30cc3b49f76a

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-889e30cc3b49f76a

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-f4f0736bd14a35db root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-b445a5ff4f115ce0 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (6)

Equal up to content, sign, reversal and a small index shift.

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