A111049: Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
A111049 ↗ nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernelinterlacing (weak)
Recurrence from A111049
\(\left(-2 + n\right)\,P_{n}(x) = \left(-6 - 6\,x + 3\,n + 3\,n x\right)\,P_{n-1}(x) + \left(4 + 8\,x + 4\,x^{2} - 2\,n - 4\,n x - 2\,n x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)
\(P_{2}(x) = 1 + 3\,x + 2\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 1 | 2 | ||||||
| 2 | 1 | 3 | 2 | 6 | |||||
| 3 | 1 | 6 | 9 | 4 | 20 | ||||
| 4 | 1 | 11 | 27 | 25 | 8 | 72 | |||
| 5 | 1 | 20 | 70 | 100 | 65 | 16 | 272 | ||
| 6 | 1 | 37 | 170 | 330 | 325 | 161 | 32 | 1056 | |
| 7 | 1 | 70 | 399 | 980 | 1295 | 966 | 385 | 64 | 4160 |
All 15 rows b-files: A028402 (row sums), A063376 (row sums), A111049, A161168 (row sums) ?
Compute more
?
Technical details
- Node id
n-3c5a3b5cd13d8cbc· revision 0 · JSON- Definition source
oeis_catalog:A111049- Fingerprint
f6-d713924feb068d4e
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): yes
Matches ?
Row sums ? A063376, A161168, A028402
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 1, 3, 2; 3: 1, 6, 9, 4; 4: 1, 11, 27, 25, 8; 5: 1, 20, 70, 100, 65, 16; 6: 1, 37, 170, 330, 325, 161, 32; 7: 1, 70, 399, 980, 1295, 966, 385, 64; 8: 1, 135, 917, 2723, 4515, 4501, 2695, 897, 128; 9: 1, 264, 2076, 7224, 14406, 17976, 14364, 7176, 2049, 256;
%F FormulaThe row polynomials satisfy (n-2)*P(n,x) = (3*n-6+(3*n-6)*x)*P(n-1,x) - (2*n-4+(4*n-8)*x+(2*n-4)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 3*x + 2*x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 3*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 6 + (3*n - 6)*x)*P[n - 1, x] - (2*n - 4 + (4*n - 8)*x + (2*n - 4)*x^2)*P[n - 2, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 3*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((3*n - 6 + (3*n - 6)*x)*v[i - 1] - (2*n - 4 + (4*n - 8)*x + (2*n - 4)*x^2)*v[i - 2])/(n - 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
A111049 triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 1, 3, 2; 3: 1, 6, 9, 4; 4: 1, 11, 27, 25, 8; 5: 1, 20, 70, 100, 65, 16; 6: 1, 37, 170, 330, 325, 161, 32; 7: 1, 70, 399, 980, 1295, 966, 385, 64; 8: 1, 135, 917, 2723, 4515, 4501, 2695, 897, 128; 9: 1, 264, 2076, 7224, 14406, 17976, 14364, 7176, 2049, 256;
%F FormulaThe row polynomials satisfy (n-2)*P(n,x) = (3*n-6+(3*n-6)*x)*P(n-1,x) - (2*n-4+(4*n-8)*x+(2*n-4)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 3*x + 2*x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 3*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 6 + (3*n - 6)*x)*P[n - 1, x] - (2*n - 4 + (4*n - 8)*x + (2*n - 4)*x^2)*P[n - 2, x])/(n - 2)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 3*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((3*n - 6 + (3*n - 6)*x)*v[i - 1] - (2*n - 4 + (4*n - 8)*x + (2*n - 4)*x^2)*v[i - 2])/(n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)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
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
Proof chains ? JSON
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