A034807: Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.
A034807 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernelinterlacing (weak)basis map preserves real roots
Recurrence from A034807
\(P_{n}(x) = \left(1\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 2 | 2 | |||
| 1 | 1 | 1 | |||
| 2 | 1 | 2 | 3 | ||
| 3 | 1 | 3 | 4 | ||
| 4 | 1 | 4 | 2 | 7 | |
| 5 | 1 | 5 | 5 | 11 | |
| 6 | 1 | 6 | 9 | 2 | 18 |
| 7 | 1 | 7 | 14 | 7 | 29 |
All 15 rows b-files: A000032 (row sums), A000204 (row sums), A034807 ?
Compute more
?
Technical details
- Node id
n-fe6a74fcc4c3def7· revision 0 · JSON- Definition source
oeis_catalog:A034807- Fingerprint
f6-088313a0be276231
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 ? A254729, A329723, A000204, A080023, A000032, A222333, A358995, A169985
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 2; 1: 1; 2: 1, 2; 3: 1, 3; 4: 1, 4, 2; 5: 1, 5, 5; 6: 1, 6, 9, 2; 7: 1, 7, 14, 7; 8: 1, 8, 20, 16, 2; 9: 1, 9, 27, 30, 9;
%F FormulaThe row polynomials satisfy P(n,x) = P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 2, P(1,x) = 1.
%t MathematicaP[0, x_] := 2;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + 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] = 2; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = v[i - 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
A034807 triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 2; 1: 1; 2: 1, 2; 3: 1, 3; 4: 1, 4, 2; 5: 1, 5, 5; 6: 1, 6, 9, 2; 7: 1, 7, 14, 7; 8: 1, 8, 20, 16, 2; 9: 1, 9, 27, 30, 9;
%F FormulaThe row polynomials satisfy P(n,x) = P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 2, P(1,x) = 1.
%t MathematicaP[0, x_] := 2;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + 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] = 2; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = v[i - 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
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
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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 (−∞, 0]: every PF input has a real-rooted image with every root <= -c/b = 0
- 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
- shift_x_plus_one of Triangle T(n,k) of coefficients of Lucas (or Cardan) polyno… Image under a basis map u^k -> P_k (A007318)
- basis_map_A013611 of Triangle T(n,k) of coefficients of Lucas (or Cardan) polyn… Image under a basis map u^k -> P_k (A013611)
- basis_map_A013610 of Triangle T(n,k) of coefficients of Lucas (or Cardan) polyn… Image under a basis map u^k -> P_k (A013610)
- basis_map_A008277 of Triangle T(n,k) of coefficients of Lucas (or Cardan) polyn… Image under a basis map u^k -> P_k (A008277)
- basis_map_A013609 of Triangle T(n,k) of coefficients of Lucas (or Cardan) polyn… Image under a basis map u^k -> P_k (A013609)
- shift_x_plus_one of Triangle T(n,k) of coefficients of Lucas (or Cardan) polyno… Image under a real-rootedness preserver (affine, shift 0)
- derivative of Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials. Image under a real-rootedness preserver (derivative polynomial, shift 0)
- theta_plus_one_squared of Triangle T(n,k) of coefficients of Lucas (or Cardan) … Image under a real-rootedness preserver (euler polynomial, shift 0)
2 more
- one_plus_derivative of Triangle T(n,k) of coefficients of Lucas (or Cardan) pol… Image under a real-rootedness preserver (derivative polynomial, shift 0)
- theta_plus_one of Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomi… Image under a real-rootedness preserver (euler polynomial, shift 0)
Related ?
Same fingerprint ?
- basis-map coefficients of A113214, shifted → Veronese 2 → basis A013611
- A178820 → Veronese 2 → basis A013611
- A178821 → Veronese 2 → basis A013611
- A113214: Riordan array (1+2x,x(1+x)).
- A178822 → Veronese 2 → basis A013611
- A034839 → basis A013611
- A228955 → basis A013611
- A038208 → Veronese 2 → basis A013611
- A119462 → basis A013611
- A028326 → Veronese 2 → basis A013611
- reversal of Riordan array (1+2x,x(1+x)).
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