A030528 → x ↦ x−15

Proved for all n: all roots real, in (11, 15] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-15 + x\right)\,P_{n-1}(x) + \left(-15 + x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -14 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-14	1							-13
2	195	-28	1						168
3	-2715	586	-42	1					-2170
4	37800	-10890	1173	-56	1				28028
5	-526275	189645	-27269	1956	-70	1			-362012
6	7327125	-3169800	570195	-54596	2935	-84	1		4675776
7	-102012750	51503175	-11124045	1332526	-95615	4110	-98	1	-60392696

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-6e46553533052aa6 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A030528
Fingerprint: f6-ff57318df0a753b8

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): no
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:          -14,           1;
  2:          195,         -28,           1;
  3:        -2715,         586,         -42,         1;
  4:        37800,      -10890,        1173,       -56,         1;
  5:      -526275,      189645,      -27269,      1956,       -70,       1;
  6:      7327125,    -3169800,      570195,    -54596,      2935,     -84,       1;
  7:   -102012750,    51503175,   -11124045,   1332526,    -95615,    4110,     -98,    1;
  8:   1420284375,  -819686250,   206641125, -29722800,   2668130, -153070,    5481, -112,    1;
  9: -19774074375, 12841017750, -3700939275, 621371190, -66977999, 4806915, -229705, 7048, -126, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(15-x)*P(n-1,x) - (15-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -14 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -14 + x;
P[n_, x_] := P[n, x] = Expand[-(15 - x)*P[n - 1, x] - (15 - x)*P[n - 2, x]];
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] = -14 + x; for(n = 2, N, my(i = n + 1); v[i] = -(15 - x)*v[i - 1] - (15 - x)*v[i - 2]); 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 (11, 15]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-8f58f94d7421b40d
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n(-x) 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-8f58f94d7421b40d
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-8f58f94d7421b40d

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

preserver_image affine, shift 0 e-8f58f94d7421b40d

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
  - theorem_check ma_wang_interval e-f811a014585c6352 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-361937c2693b166b basis_map (A027467) → Triangle read by rows: T(n,k) = binomial(k,n-k).

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