A128099 → basis A013609 → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-4 + 2\,n\right)\,P_{n}(x) = \left(-26 - 48\,x + 10\,n + 16\,n x\right)\,P_{n-1}(x) + \left(-9 - 34\,x - 32\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 4\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | |||||||
| 1 | 0 | |||||||
| 2 | 4 | 4 | ||||||
| 3 | 8 | 8 | ||||||
| 4 | 28 | 32 | 60 | |||||
| 5 | 64 | 96 | 160 | |||||
| 6 | 164 | 384 | 192 | 740 | ||||
| 7 | 376 | 1088 | 768 | 2232 | ||||
| 8 | 876 | 3168 | 3456 | 1024 | 8524 | |||
| 9 | 1968 | 8352 | 11520 | 5120 | 26960 | |||
| 10 | 4404 | 21696 | 37440 | 25600 | 5120 | 94260 | ||
| 11 | 9704 | 54144 | 110592 | 97280 | 30720 | 302440 | ||
| 12 | 21244 | 132768 | 315648 | 348160 | 168960 | 24576 | 1011356 | |
| 13 | 46112 | 318688 | 861696 | 1132544 | 716800 | 172032 | 3247872 | |
| 14 | 99524 | 754176 | 2289600 | 3512320 | 2795520 | 1032192 | 114688 | 10598020 |
Compute more
?
Technical details
- Node id
n-7e3ff7af8de21331· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-05f67b4a28449ab6
Observed on the stored rows ?
- Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
- 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: 0; 2: 4; 3: 8; 4: 28, 32; 5: 64, 96; 6: 164, 384, 192; 7: 376, 1088, 768; 8: 876, 3168, 3456, 1024; 9: 1968, 8352, 11520, 5120;
%F FormulaThe row polynomials satisfy (2*n-4)*P(n,x) = (10*n-26+(16*n-48)*x)*P(n-1,x) - (9+34*x+32*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 4.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((10*n - 26 + (16*n - 48)*x)*P[n - 1, x] - (9 + 34*x + 32*x^2)*D[P[n - 1, x], x])/(2*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 4; for(n = 3, N, my(i = n + 1); v[i] = ((10*n - 26 + (16*n - 48)*x)*v[i - 1] - (9 + 34*x + 32*x^2)*deriv(v[i - 1], x))/(2*n - 4)); 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]
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Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
basis_map_A013609 of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X…?
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Geometric basis map (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pi…?
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Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
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from
Pascal-(1,2,1) array read by antidiagonals.?
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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 ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
Pascal-(1,2,1) array read by antidiagonals.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
A Jacobsthal triangle.?
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Geometric basis map (elementary) polytool ? step ›
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from
basis-map coefficients of A Jacobsthal triangle.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
A Jacobsthal triangle.?
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Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
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from
A Jacobsthal triangle.?
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from
basis-map coefficients of A Jacobsthal triangle.?
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pi…?
- ↑ Parent-row provers of polytool (above)
- ↑ Gamma criterion (above)
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
A Jacobsthal triangle.?
- ↑ 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)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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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
A Jacobsthal triangle.?
-
from
Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pi…?
- ↑ 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)
- ↑ 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)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
-
-
from
A Jacobsthal triangle.?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Parent-row provers of polytool (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
-
from
Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pi…?
-
Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
-
from
Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pie…?
-
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…?
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Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
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from
delannoy?
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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, reversal and Möbius substitutions (elementary) polytool ? step ›
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
Geometric basis map (elementary) polytool ? step ›
-
-
from
Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pie…?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pie…?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
-
from
Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pie…?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pie…?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
-
from
Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pie…?
-
-
from
basis_map_A013609 of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X…?
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.