A116412 → basis A007318 → basis A007318
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(2 + x\right)\,P_{n-2}(x) + \left(4 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 5 | 1 | 6 | ||||||
| 2 | 22 | 10 | 1 | 33 | |||||
| 3 | 98 | 69 | 15 | 1 | 183 | ||||
| 4 | 436 | 416 | 141 | 20 | 1 | 1014 | |||
| 5 | 1940 | 2336 | 1079 | 238 | 25 | 1 | 5619 | ||
| 6 | 8632 | 12552 | 7350 | 2212 | 360 | 30 | 1 | 31137 | |
| 7 | 38408 | 65452 | 46446 | 17753 | 3940 | 507 | 35 | 1 | 172542 |
Compute more
?
Technical details
- Node id
n-c0d4619303a87a33· revision 0 · JSON- Definition source
derived:basis_map_A007318- Fingerprint
f6-8093bc57471dc962
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): no
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 5, 1; 2: 22, 10, 1; 3: 98, 69, 15, 1; 4: 436, 416, 141, 20, 1; 5: 1940, 2336, 1079, 238, 25, 1; 6: 8632, 12552, 7350, 2212, 360, 30, 1; 7: 38408, 65452, 46446, 17753, 3940, 507, 35, 1; 8: 170896, 333952, 278488, 129232, 36445, 6388, 679, 40, 1; 9: 760400, 1676016, 1606248, 877368, 300645, 66951, 9681, 876, 45, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (2+x)*P(n-2,x) + (4+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 5 + x;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 2, x] + (4 + x)*P[n - 1, 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] = 5 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + x)*v[i - 2] + (4 + x)*v[i - 1]); 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 Geometric basis map (elementary)
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
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Geometric basis map (elementary) polytool ? step ›
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from
basis_map_A007318 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
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Geometric basis map (elementary) polytool ? step ›
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from
Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Liu–Wang theorem with a lag row (above)
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
-
from
Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
-
-
from
basis_map_A007318 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).?
Related ?
Candidate equalities ?
- A116412 → basis A007318 → x ↦ x+1 15 rows; no all-index equality certificate was found
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