A208509: Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.
A208509 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernelinterlacing (weak)basis map preserves real roots
Recurrence from A208509
\(P_{n}(x) = \left(2\right)\,P_{n-1}(x) + \left(-1 + x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 1 | |||
| 1 | 3 | 3 | |||
| 2 | 5 | 1 | 6 | ||
| 3 | 7 | 5 | 12 | ||
| 4 | 9 | 14 | 1 | 24 | |
| 5 | 11 | 30 | 7 | 48 | |
| 6 | 13 | 55 | 27 | 1 | 96 |
| 7 | 15 | 91 | 77 | 9 | 192 |
All 15 rows b-files: A003945 (row sums), A208509 ?
Compute more
?
Technical details
- Node id
n-d4c5acf24bc40277· revision 0 · JSON- Definition source
oeis_catalog:A208509- Fingerprint
f6-930ba27ff87052d7
Observed on the stored rows ?
- Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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
Matches ?
Row sums ? A122391, A003945, A167881, A168728, A168680, A168824, A167648, A168776
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 3; 2: 5, 1; 3: 7, 5; 4: 9, 14, 1; 5: 11, 30, 7; 6: 13, 55, 27, 1; 7: 15, 91, 77, 9; 8: 17, 140, 182, 44, 1; 9: 19, 204, 378, 156, 11;
%F FormulaThe row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*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; for(n = 2, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - 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)polytool c48629374597
A208509 triangle, offset 1, from P0
%e ExampleTriangle begins: 1: 1; 2: 3; 3: 5, 1; 4: 7, 5; 5: 9, 14, 1; 6: 11, 30, 7; 7: 13, 55, 27, 1; 8: 15, 91, 77, 9; 9: 17, 140, 182, 44, 1; 10: 19, 204, 378, 156, 11;
%F FormulaThe row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3; for(n = 3, N, my(i = n); v[i] = 2*v[i - 1] - (1 - 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, 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
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
- Basis map preserves real-rootedness? Basis map u^k -> P_k sends roots in (−∞, 0] to roots in (−∞, 1]: every PF input has a real-rooted image with every root <= -c/b = 1
- Basis map preserves real-rootedness? Basis map u^k -> P_k sends roots in (−∞, 0] to roots in (−∞, 0]: every PF input has a PF image (zero allowed)
Proof chains ? JSON
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Constant-current basis map [Bra15, Lemma 8.5 of arXiv:1410.6601] polytool RealRooted orderTwoBasis_pf_equivalences RealRooted basisTransform_orderTwoBasis_preserves_bounded_pf_iff ? step › direct check
Used in 10
- basis_map_A013611 of Triangle of coefficients of polynomials v(n,x) jointly gen… Image under a basis map u^k -> P_k (A013611)
- basis_map_A008277 of Triangle of coefficients of polynomials v(n,x) jointly gen… Image under a basis map u^k -> P_k (A008277)
- basis_map_A007318 of Triangle of coefficients of polynomials v(n,x) jointly gen… Image under a basis map u^k -> P_k (A007318)
- basis_map_A013609 of Triangle of coefficients of polynomials v(n,x) jointly gen… Image under a basis map u^k -> P_k (A013609)
- basis_map_A013610 of Triangle of coefficients of polynomials v(n,x) jointly gen… Image under a basis map u^k -> P_k (A013610)
- theta_plus_one of Triangle of coefficients of polynomials v(n,x) jointly genera… Image under a real-rootedness preserver (euler polynomial, shift 0)
- shift_x_plus_one of Triangle of coefficients of polynomials v(n,x) jointly gene… Image under a real-rootedness preserver (affine, shift 0)
- one_plus_derivative of Triangle of coefficients of polynomials v(n,x) jointly g… Image under a real-rootedness preserver (derivative polynomial, shift 0)
2 more
- derivative of Triangle of coefficients of polynomials v(n,x) jointly generated … Image under a real-rootedness preserver (derivative polynomial, shift 0)
- theta_plus_one_squared of Triangle of coefficients of polynomials v(n,x) jointl… Image under a real-rootedness preserver (euler polynomial, shift 0)
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [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