A038226 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(11 + 8\,x\right)\,P_{n-1}(x) + \left(8\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 121 + 240\,x + 64\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	121	240	64								425
1	1331	5528	3648	512							11019
2	14641	115680	142720	47104	4096						324241
3	161051	2315048	4778880	2790400	552960	32768					10631107
4	1771561	45274320	147550144	135895040	46100480	6094848	262144				382948537
5	19487171	874384568	4346048448	5936727552	3069480960	679641088	64225280	2097152			14992092219
6	214358881	16769204160	124338384640	242553851904	179481501696	59217543168	9226420224	654311424	16777216		632472353313
7	2357947691	320329750088	3491290018560	9484091893760	9658135388160	4455946715136	1018099138560	117650227200	6492782592	134217728	28554528079475

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-169167e70da5443c · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-b1ddbb9f02bc0f91

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:          121,             240,               64;
  1:         1331,            5528,             3648,               512;
  2:        14641,          115680,           142720,             47104,              4096;
  3:       161051,         2315048,          4778880,           2790400,            552960,             32768;
  4:      1771561,        45274320,        147550144,         135895040,          46100480,           6094848,           262144;
  5:     19487171,       874384568,       4346048448,        5936727552,        3069480960,         679641088,         64225280,          2097152;
  6:    214358881,     16769204160,     124338384640,      242553851904,      179481501696,       59217543168,       9226420224,        654311424,        16777216;
  7:   2357947691,    320329750088,    3491290018560,     9484091893760,     9658135388160,     4455946715136,    1018099138560,     117650227200,      6492782592,      134217728;
  8:  25937424601,   6105128833200,   96827468501824,   359873536430080,   491172556840960,   304518365577216,   95715422896128,   16027358330880,   1428160512000,    63082332160,   1073741824;
  9: 285311670611, 116204947227608, 2663182680214848, 13370193523067392, 23999408235601920, 19459817099165696, 8083356875489280, 1839556391337984, 235330905047040, 16661117665280, 602369163264, 8589934592;

%F Formula

The row polynomials satisfy P(n,x) = (11+8*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 121 + 240*x + 64*x^2.

%t Mathematica

P[0, x_] := 121 + 240*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 8*x)*P[n - 1, x] + 8*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] = 121 + 240*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 8*x)*v[i - 1] + 8*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-fc0676c2630bb222
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-fc0676c2630bb222
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-fc0676c2630bb222

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-fc0676c2630bb222

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-fdf6b848b46b3343 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-711ca5eff5ba56ea 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