A038331 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(17 + 5\,x\right)\,P_{n-1}(x) + \left(5\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 289 + 195\,x + 25\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	289	195	25								509
1	4913	5735	1650	125							12423
2	83521	150735	73225	12250	625						320356
3	1419857	3733775	2730750	758125	84375	3125					8730007
4	24137569	89242335	92399125	37913750	6912500	553125	15625				251174029
5	410338673	2084019215	2940988050	1675235625	445331250	57793750	3500000	78125			7617284688
6	6975757441	47900116095	89826773425	68312480250	24853434375	4653993750	453468750	21562500	390625		242997977211
7	118587876497	1088681341295	2664823462950	2635133235125	1261139473125	319734909375	44583000000	3388593750	130078125	1953125	8136203923367

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-ba23d827179717e1 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-751f1aae2d0ee2d6

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): no

OEIS matches ?

No OEIS matches.

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:            289,             195,               25;
  1:           4913,            5735,             1650,              125;
  2:          83521,          150735,            73225,            12250,              625;
  3:        1419857,         3733775,          2730750,           758125,            84375,             3125;
  4:       24137569,        89242335,         92399125,         37913750,          6912500,           553125,           15625;
  5:      410338673,      2084019215,       2940988050,       1675235625,        445331250,         57793750,         3500000,          78125;
  6:     6975757441,     47900116095,      89826773425,      68312480250,      24853434375,       4653993750,       453468750,       21562500,        390625;
  7:   118587876497,   1088681341295,    2664823462950,    2635133235125,    1261139473125,     319734909375,     44583000000,     3388593750,     130078125,      1953125;
  8:  2015993900449,  24543928890975,   77393640206125,   97648380838750,   59837826681250,   19734563559375,   3694075546875,   399121875000,   24357421875,    771484375,    9765625;
  9: 34271896307633, 550046405103695, 2212347930020250, 3511716387870625, 2702241491400000, 1128040802900000, 272294368500000, 39224715234375, 3383982421875, 169619140625, 4511718750, 48828125;

%F Formula

The row polynomials satisfy P(n,x) = (17+5*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 289 + 195*x + 25*x^2.

%t Mathematica

P[0, x_] := 289 + 195*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(17 + 5*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 289 + 195*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 5*x)*v[i - 1] + 5*x*deriv(v[i - 1], x)); 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-fd4b406e93b80ec2
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-fd4b406e93b80ec2
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-fd4b406e93b80ec2

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 A008277 e-fd4b406e93b80ec2

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-49423b91d6bf9513 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-b8d4501ecbbc70e3 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741