A152842 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + 13\,x + 3\,x^{2}\right)\,P_{n-2}(x) + \left(13\,x + 6\,x^{2}\right)\,P_{n-2}'(x) + \left(3\,x^{2}\right)\,P_{n-2}''(x)\)

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

\(P_{1}(x) = 4 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	3							7
2	8	13	3						24
3	32	115	69	9					225
4	64	377	391	114	9				955
5	256	2831	5041	2655	486	27			11296
6	512	8749	22995	18316	5571	675	27		56845
7	2048	62779	256197	307093	144084	29538	2619	81	804439

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-9bbaadfa7220e0e4 · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-5bf85aaad7b429ab

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:     4,       3;
  2:     8,      13,        3;
  3:    32,     115,       69,        9;
  4:    64,     377,      391,      114,        9;
  5:   256,    2831,     5041,     2655,      486,       27;
  6:   512,    8749,    22995,    18316,     5571,      675,      27;
  7:  2048,   62779,   256197,   307093,   144084,    29538,    2619,     81;
  8:  4096,  190385,  1087567,  1791662,  1171597,   350850,   50490,   3348,    81;
  9: 16384, 1344983, 11446825, 26554307, 24120538, 10180941, 2163330, 235170, 12312, 243;

%F Formula

The row polynomials satisfy P(n,x) = (8+13*x+3*x^2)*P(n-2,x) + (13*x+6*x^2)*d/dx P(n-2,x) + 3*x^2*d^2/dx^2 P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[(8 + 13*x + 3*x^2)*P[n - 2, x] + (13*x + 6*x^2)*D[P[n - 2, x], x] + 3*x^2*D[P[n - 2, 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; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 13*x + 3*x^2)*v[i - 2] + (13*x + 6*x^2)*deriv(v[i - 2], x) + 3*x^2*deriv(deriv(v[i - 2], 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-79898a409f2d103b
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-79898a409f2d103b
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-79898a409f2d103b

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-79898a409f2d103b

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DEL…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-596b986e3a00db44 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