A038336 → basis A007318

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(22 + 10\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 484 + 440\,x + 100\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	484	440	100								1024
1	10648	14520	6600	1000							32768
2	234256	425920	290400	88000	10000						1048576
3	5153632	11712800	10648000	4840000	1100000	100000					33554432
4	113379904	309217920	351384000	212960000	72600000	13200000	1000000				1073741824
5	2494357888	7936593280	10822627200	8198960000	3726800000	1016400000	154000000	10000000			34359738368
6	54875873536	199548631040	317463731200	288603392000	163979200000	59628800000	13552000000	1760000000	100000000		1099511627776
7	1207269217792	4938828618240	8979688396800	9523911936000	6493576320000	2951625600000	894432000000	174240000000	19800000000	1000000000	35184372088832

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-a11e46881092378a · revision 1 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-2235b7dd30e0ad6a

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 ?

A009976: 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:             484,              440,              100;
  1:           10648,            14520,             6600,             1000;
  2:          234256,           425920,           290400,            88000,            10000;
  3:         5153632,         11712800,         10648000,          4840000,          1100000,           100000;
  4:       113379904,        309217920,        351384000,        212960000,         72600000,         13200000,          1000000;
  5:      2494357888,       7936593280,      10822627200,       8198960000,       3726800000,       1016400000,        154000000,        10000000;
  6:     54875873536,     199548631040,     317463731200,     288603392000,     163979200000,      59628800000,      13552000000,      1760000000,       100000000;
  7:   1207269217792,    4938828618240,    8979688396800,    9523911936000,    6493576320000,    2951625600000,     894432000000,    174240000000,     19800000000,     1000000000;
  8:  26559922791424,  120726921779200,  246941430912000,  299322946560000,  238097798400000,  129871526400000,   49193760000000,  12777600000000,   2178000000000,   220000000000,   10000000000;
  9: 584318301411328, 2921591507056640, 6639980697856000, 9054519133440000, 8231381030400000, 5238151564800000, 2380977984000000, 773044800000000, 175692000000000, 26620000000000, 2420000000000, 100000000000;

%F Formula

The row polynomials satisfy P(n,x) = (22+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 440*x + 100*x^2.

%t Mathematica

P[0, x_] := 484 + 440*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 10*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] = 484 + 440*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 10*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-344e4e749f34f1c6
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-344e4e749f34f1c6
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-344e4e749f34f1c6

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 A007318 e-344e4e749f34f1c6

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(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-2f1e093ae220e593 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-3f0a5f7ceded39cb 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