A272867 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + 7\,x + x^{2}\right)\,P_{n-1}(x) + \left(7\,x + 2\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	7	1													14
2	36	133	91	18	1											279
3	216	1981	3235	1728	371	33	1									7565
4	1296	27265	89215	94590	42765	9306	1010	52	1							265500
5	7776	363517	2209411	4130100	3286075	1303731	279747	33540	2220	75	1					11616193
6	46656	4780153	51886471	160960338	207060161	132948816	47269768	9872304	1246047	95175	4261	102	1			616170253
7	279936	62468581	1184570395	5887222488	11669138271	11376740529	6154231937	1988687896	401621698	51951060	4332748	229908	7441	133	1	38781483022

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-19b5be97f4a24d38 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-4e1e31e01a4728f2

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
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:        1;
  1:        6,           7,            1;
  2:       36,         133,           91,            18,              1;
  3:      216,        1981,         3235,          1728,            371,             33,              1;
  4:     1296,       27265,        89215,         94590,          42765,           9306,           1010,             52,              1;
  5:     7776,      363517,      2209411,       4130100,        3286075,        1303731,         279747,          33540,           2220,            75,            1;
  6:    46656,     4780153,     51886471,     160960338,      207060161,      132948816,       47269768,        9872304,        1246047,         95175,         4261,          102,           1;
  7:   279936,    62468581,   1184570395,    5887222488,    11669138271,    11376740529,     6154231937,     1988687896,      401621698,      51951060,      4332748,       229908,        7441,       133,        1;
  8:  1679616,   814051105,  26623045855,  207371085030,   614498823205,   874905468822,   685103818350,   321209873244,    95308038117,   18551086410,   2419634366,    213307500,    12651982,    494228,    12116,    168,     1;
  9: 10077696, 10594421677, 593035148371, 7136913361500, 30989393018675, 62794087031247, 68925989624479, 45049235722140, 18676859654280, 5128997933745, 960744503123, 125049561792, 11414291870, 730558626, 32426682, 972216, 18690, 207, 1;

%F Formula

The row polynomials satisfy P(n,x) = (6+7*x+x^2)*P(n-1,x) + (7*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(6 + 7*x + x^2)*P[n - 1, x] + (7*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (6 + 7*x + x^2)*v[i - 1] + (7*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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-efe78907c9beaa2c
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-efe78907c9beaa2c
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-efe78907c9beaa2c

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-efe78907c9beaa2c

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-eab26daf7024ea74 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[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