A038284 → 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

\(P_{n}(x) = \left(14 + 24\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 196 + 672\,x + 576\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	196	672	576								1444
1	2744	14112	24192	13824							54872
2	38416	263424	677376	774144	331776						2085136
3	537824	4609920	15805440	27095040	23224320	7962624					79235168
4	7529536	77446656	331914240	758661120	975421440	668860416	191102976				3010936384
5	105413504	1264962048	6505519104	18587197440	31863767040	32774160384	18728091648	4586471424			114415582592
6	1475789056	20239392768	121436356608	416353222656	892185477120	1223568654336	1048773132288	513684799488	110075314176		4347792138496
7	20661046784	318770436096	2185854418944	8743417675776	22483074023424	38542412611584	44048471556096	32362142367744	13869489586176	2641807540224	165216101262848

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-0b6560aa9867a4e4 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-48a0cbb267bcb941

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 ?

A009982: 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:           196,            672,             576;
  1:          2744,          14112,           24192,            13824;
  2:         38416,         263424,          677376,           774144,            331776;
  3:        537824,        4609920,        15805440,         27095040,          23224320,           7962624;
  4:       7529536,       77446656,       331914240,        758661120,         975421440,         668860416,         191102976;
  5:     105413504,     1264962048,      6505519104,      18587197440,       31863767040,       32774160384,       18728091648,        4586471424;
  6:    1475789056,    20239392768,    121436356608,     416353222656,      892185477120,     1223568654336,     1048773132288,      513684799488,      110075314176;
  7:   20661046784,   318770436096,   2185854418944,    8743417675776,    22483074023424,    38542412611584,    44048471556096,    32362142367744,    13869489586176,     2641807540224;
  8:  289254654976,  4958651228160,  38252452331520,  174868353515520,   524605060546560,  1079187553124352,  1541696504463360,  1510233310494720,   970864271032320,   369853055631360,   63403380965376;
  9: 4049565169664, 76363228913664, 654541962117120, 3366215805173760, 11541311332024320, 27699147196858368, 47484252337471488, 58143982454046720, 49837699246325760, 28478685283614720, 9764120668667904, 1521681143169024;

%F Formula

The row polynomials satisfy P(n,x) = (14+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 672*x + 576*x^2.

%t Mathematica

P[0, x_] := 196 + 672*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(14 + 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] = 196 + 672*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 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-06104b6211fa5aaa
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-06104b6211fa5aaa
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-06104b6211fa5aaa

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-06104b6211fa5aaa

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-4ef268212a4f2f46 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-4fdd1b3030c1c25c root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (3)

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