A131436: Triangle read by rows, (n-1) zeros followed by 2^n - 1.
A131436 ↗ palindromicγ-positivenonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence from A131436
\(P_{n}(x) = \left(3\,x\right)\,P_{n-1}(x) + \left(-2\,x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 0 | 3 | 3 | ||||||
| 2 | 0 | 0 | 7 | 7 | |||||
| 3 | 0 | 0 | 0 | 15 | 15 | ||||
| 4 | 0 | 0 | 0 | 0 | 31 | 31 | |||
| 5 | 0 | 0 | 0 | 0 | 0 | 63 | 63 | ||
| 6 | 0 | 0 | 0 | 0 | 0 | 0 | 127 | 127 | |
| 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 255 | 255 |
All 15 rows b-files: A126646 (row sums), A131436, A168604 (row sums) ?
Compute more
?
Technical details
- Node id
n-6394a556f7ad3cd4· revision 0 · JSON- Definition source
oeis_catalog:A131436- Fingerprint
f6-edf6954a4cd8adfc
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: yes
- γ-positive (gamma_positive): yes
- Nonnegative coefficients (
nonn): yes - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes
Matches ?
Row sums ? A255047, A126646, A153010, A168604, A239678, A088041, A060152, A000225
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 0, 3; 2: 0, 0, 7; 3: 0, 0, 0, 15; 4: 0, 0, 0, 0, 31; 5: 0, 0, 0, 0, 0, 63; 6: 0, 0, 0, 0, 0, 0, 127; 7: 0, 0, 0, 0, 0, 0, 0, 255; 8: 0, 0, 0, 0, 0, 0, 0, 0, 511; 9: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1023;
%F FormulaThe row polynomials satisfy P(n,x) = 3*x*P(n-1,x) - 2*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 3*x;
P[n_, x_] := P[n, x] = Expand[3*x*P[n - 1, x] - 2*x^2*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3*x; for(n = 2, N, my(i = n + 1); v[i] = 3*x*v[i - 1] - 2*x^2*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)polytool c48629374597
A131436 triangle, offset 1, from P0
%e ExampleTriangle begins: 1: 1; 2: 0, 3; 3: 0, 0, 7; 4: 0, 0, 0, 15; 5: 0, 0, 0, 0, 31; 6: 0, 0, 0, 0, 0, 63; 7: 0, 0, 0, 0, 0, 0, 127; 8: 0, 0, 0, 0, 0, 0, 0, 255; 9: 0, 0, 0, 0, 0, 0, 0, 0, 511; 10: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1023;
%F FormulaThe row polynomials satisfy P(n,x) = 3*x*P(n-1,x) - 2*x^2*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 3*x;
P[n_, x_] := P[n, x] = Expand[3*x*P[n - 1, x] - 2*x^2*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3*x; for(n = 3, N, my(i = n); v[i] = 3*x*v[i - 1] - 2*x^2*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
Proof chains ? JSON
Related ?
Same fingerprint ?
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- A129184: Shift operator, right.
- A085881 → γ
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- A127773 → reverse
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- basis-map coefficients of A297191, shifted → D
- A127648 → D
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