A169803 → D → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(-4 - 4\,x + 3\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-5 - 9\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | |||||||
| 1 | 1 | 1 | ||||||
| 2 | 2 | 2 | ||||||
| 3 | 5 | 2 | 7 | |||||
| 4 | 10 | 6 | 16 | |||||
| 5 | 20 | 18 | 3 | 41 | ||||
| 6 | 38 | 44 | 12 | 94 | ||||
| 7 | 71 | 102 | 42 | 4 | 219 | |||
| 8 | 130 | 222 | 120 | 20 | 492 | |||
| 9 | 235 | 466 | 315 | 80 | 5 | 1101 | ||
| 10 | 420 | 948 | 768 | 260 | 30 | 2426 | ||
| 11 | 744 | 1884 | 1782 | 760 | 135 | 6 | 5311 | |
| 12 | 1308 | 3672 | 3972 | 2044 | 490 | 42 | 11528 | |
| 13 | 2285 | 7044 | 8580 | 5180 | 1575 | 210 | 7 | 24881 |
| 14 | 3970 | 13332 | 18060 | 12520 | 4620 | 840 | 56 | 53398 |
Compute more
?
Technical details
- Node id
n-c99a4c5c96f874a0· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-e7a207c59c800cef
Observed on the stored rows ?
- Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half affine: slope 1, intercept -1)
- 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: 0; 1: 1; 2: 2; 3: 5, 2; 4: 10, 6; 5: 20, 18, 3; 6: 38, 44, 12; 7: 71, 102, 42, 4; 8: 130, 222, 120, 20; 9: 235, 466, 315, 80, 5;
%F FormulaThe row polynomials satisfy (n-1)*P(n,x) = (3*n-4+(2*n-4)*x)*P(n-1,x) - (5+9*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 4 + (2*n - 4)*x)*P[n - 1, x] - (5 + 9*x + 4*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 4 + (2*n - 4)*x)*v[i - 1] - (5 + 9*x + 4*x^2)*deriv(v[i - 1], x))/(n - 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 (−∞, -1]
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
Proof chains ? JSON
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
derivative of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).?
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
-
Geometric basis map (elementary) polytool ? step ›
-
from
basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
-
Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
-
-
from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
-
-
from
basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).?
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…?
-
Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
-
from
delannoy?
-
Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
delannoy?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
delannoy?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
-
Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
-
Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
-
from
delannoy?
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
-
from
delannoy?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
-
-
from
delannoy?
-
from
Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…?
-
Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
- ↑ Ma–Wang interval theorem (above)
-
-
from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
-
from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
-
-
from
Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).?
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
-
from
derivative of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).?
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
-
from
shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).?
Related ?
Same fingerprint ?
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
- [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.