A038301 → 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

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

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

Rows ?

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

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-7d2bbb701a5b14c4 · 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:             400,              880,               484;
  1:            8000,            26400,             29040,             10648;
  2:          160000,           704000,           1161600,            851840,            234256;
  3:         3200000,         17600000,          38720000,          42592000,          23425600,            5153632;
  4:        64000000,        422400000,        1161600000,        1703680000,        1405536000,          618435840,          113379904;
  5:      1280000000,       9856000000,       32524800000,       59628800000,       65591680000,        43290508800,        15873186560,         2494357888;
  6:     25600000000,     225280000000,      867328000000,     1908121600000,     2623667200000,      2308827136000,      1269854924800,       399097262080,       54875873536;
  7:    512000000000,    5068800000000,    22302720000000,    57243648000000,    94452019200000,    103897221120000,     76191295488000,     35918753587200,     9877657236480,     1207269217792;
  8:  10240000000000,  112640000000000,   557568000000000,  1635532800000000,  3148400640000000,   4155888844800000,   3809564774400000,   2394583572480000,   987765723648000,   241453843558400,   26559922791424;
  9: 204800000000000, 2478080000000000, 13629440000000000, 44977152000000000, 98949734400000000, 152382590976000000, 167620850073600000, 131702096486400000, 72436153067520000, 26559922791424000, 5843183014113280, 584318301411328;

%F Formula

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

%t Mathematica

P[0, x_] := 400 + 880*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 22*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 + 880*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 22*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-21c2803b3d0372c9
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-21c2803b3d0372c9
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-21c2803b3d0372c9

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-21c2803b3d0372c9

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-c85fc2223c79991e root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-38b35fa9e37eb25d 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