A102547 → reverse → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 3\,x + x^{2}\right)\,P_{n-3}(x) + \left(3\,x + 2\,x^{2}\right)\,P_{n-3}'(x) + \left(x^{2}\right)\,P_{n-3}''(x) + \left(1 + x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 3\,x + x^{2}\)

\(P_{2}(x) = 1 + 7\,x + 6\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	1	1								2
1	1	3	1							5
2	1	7	6	1						15
3	2	22	31	11	1					67
4	3	61	140	85	17	1				307
5	4	156	571	545	185	24	1			1486
6	6	410	2260	3166	1625	351	32	1		7851
7	9	1079	8758	17325	12621	4039	605	41	1	44478

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-f57d42b6ca5bc90e · revision 1 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-5b0ec46c40c78bca

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
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;
  1:  1,    3,      1;
  2:  1,    7,      6,      1;
  3:  2,   22,     31,     11,      1;
  4:  3,   61,    140,     85,     17,      1;
  5:  4,  156,    571,    545,    185,     24,      1;
  6:  6,  410,   2260,   3166,   1625,    351,     32,     1;
  7:  9, 1079,   8758,  17325,  12621,   4039,    605,    41,    1;
  8: 13, 2803,  33276,  90931,  90531,  40299,   8820,   972,   51,  1;
  9: 19, 7277, 125027, 463886, 614965, 366002, 109795, 17491, 1480, 62, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+3*x+x^2)*P(n-3,x) + (3*x+2*x^2)*d/dx P(n-3,x) + x^2*d^2/dx^2 P(n-3,x) + (1+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1 + x, P(1,x) = 1 + 3*x + x^2, P(2,x) = 1 + 7*x + 6*x^2 + x^3.

%t Mathematica

P[0, x_] := 1 + x;
P[1, x_] := 1 + 3*x + x^2;
P[2, x_] := 1 + 7*x + 6*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x + x^2)*P[n - 3, x] + (3*x + 2*x^2)*D[P[n - 3, x], x] + x^2*D[P[n - 3, x], {x, 2}] + (1 + x)*P[n - 1, x] + 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] = 1 + x; v[2] = 1 + 3*x + x^2; v[3] = 1 + 7*x + 6*x^2 + x^3; for(n = 3, N, my(i = n + 1); v[i] = (1 + 3*x + x^2)*v[i - 3] + (3*x + 2*x^2)*deriv(v[i - 3], x) + x^2*deriv(deriv(v[i - 3], x), x) + (1 + x)*v[i - 1] + 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-535fcb4dbc81ccf5
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-535fcb4dbc81ccf5
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-535fcb4dbc81ccf5

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-535fcb4dbc81ccf5

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?
- premise (sequence): Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - preserver_image reversal, shift 0 e-0e9e334768ea0a35
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    preserver_image reversal, shift 0 e-9cfbf7a52f4ba322
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    no recorded property covers this requirement
    
    parent_provers paper_theorem e-eef18680cd27a7c9 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