A038336 → basis A013609

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(22 + 20\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 484 + 880\,x + 400\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	484	880	400								1764
1	10648	29040	26400	8000							74088
2	234256	851840	1161600	704000	160000						3111696
3	5153632	23425600	42592000	38720000	17600000	3200000					130691232
4	113379904	618435840	1405536000	1703680000	1161600000	422400000	64000000				5489031744
5	2494357888	15873186560	43290508800	65591680000	59628800000	32524800000	9856000000	1280000000			230539333248
6	54875873536	399097262080	1269854924800	2308827136000	2623667200000	1908121600000	867328000000	225280000000	25600000000		9682651996416
7	1207269217792	9877657236480	35918753587200	76191295488000	103897221120000	94452019200000	57243648000000	22302720000000	5068800000000	512000000000	406671383849472

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-c4b5ec42f7a2f0ec · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-c4264000c5d1dd22

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 ?

A009986: 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,              880,               400;
  1:           10648,            29040,             26400,              8000;
  2:          234256,           851840,           1161600,            704000,             160000;
  3:         5153632,         23425600,          42592000,          38720000,           17600000,            3200000;
  4:       113379904,        618435840,        1405536000,        1703680000,         1161600000,          422400000,           64000000;
  5:      2494357888,      15873186560,       43290508800,       65591680000,        59628800000,        32524800000,         9856000000,        1280000000;
  6:     54875873536,     399097262080,     1269854924800,     2308827136000,      2623667200000,      1908121600000,       867328000000,      225280000000,       25600000000;
  7:   1207269217792,    9877657236480,    35918753587200,    76191295488000,    103897221120000,     94452019200000,     57243648000000,    22302720000000,     5068800000000,      512000000000;
  8:  26559922791424,  241453843558400,   987765723648000,  2394583572480000,   3809564774400000,   4155888844800000,   3148400640000000,  1635532800000000,   557568000000000,   112640000000000,   10240000000000;
  9: 584318301411328, 5843183014113280, 26559922791424000, 72436153067520000, 131702096486400000, 167620850073600000, 152382590976000000, 98949734400000000, 44977152000000000, 13629440000000000, 2478080000000000, 204800000000000;

%F Formula

The row polynomials satisfy P(n,x) = (22+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 880*x + 400*x^2.

%t Mathematica

P[0, x_] := 484 + 880*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 20*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 + 880*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 20*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-0c8cc5263f914382
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-0c8cc5263f914382
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-0c8cc5263f914382

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 A013609 e-0c8cc5263f914382

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 (2)

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