A113953 → basis A007318 → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(1 + x - 3\,n - n x + 2\,n^{2}\right)\,P_{n}(x) = \left(-40 - 18\,x - 2\,x^{2} + 42\,n + 10\,n x - 8\,n^{2}\right)\,P_{n-1}(x) + \left(-27 - 39\,x - 13\,x^{2} - x^{3} + 18\,n + 20\,n x + 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | ||||||||||||||
| 1 | 1 | 1 | |||||||||||||
| 2 | 4 | 2 | 6 | ||||||||||||
| 3 | 11 | 14 | 3 | 28 | |||||||||||
| 4 | 30 | 56 | 30 | 4 | 120 | ||||||||||
| 5 | 73 | 188 | 162 | 52 | 5 | 480 | |||||||||
| 6 | 176 | 566 | 672 | 356 | 80 | 6 | 1856 | ||||||||
| 7 | 407 | 1586 | 2409 | 1788 | 665 | 114 | 7 | 6976 | |||||||
| 8 | 930 | 4236 | 7830 | 7504 | 3950 | 1116 | 154 | 8 | 25728 | ||||||
| 9 | 2085 | 10896 | 23760 | 27944 | 19130 | 7680 | 1736 | 200 | 9 | 93440 | |||||
| 10 | 4628 | 27258 | 68472 | 95512 | 80720 | 42348 | 13608 | 2552 | 252 | 10 | 335360 | ||||
| 11 | 10163 | 66646 | 189567 | 306056 | 308230 | 200484 | 84406 | 22472 | 3591 | 310 | 11 | 1191936 | |||
| 12 | 22150 | 160000 | 508254 | 932428 | 1091020 | 848784 | 444332 | 155144 | 35118 | 4880 | 374 | 12 | 4202496 | ||
| 13 | 47937 | 378244 | 1327326 | 2727724 | 3638155 | 3298728 | 2071188 | 900824 | 267399 | 52500 | 6446 | 444 | 13 | 14716928 | |
| 14 | 103160 | 882718 | 3391200 | 7717692 | 11560920 | 11980530 | 8788864 | 4593800 | 1700136 | 437410 | 75680 | 8316 | 520 | 14 | 51240960 |
Compute more
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Technical details
- Node id
n-258d9f1a1489b4ba· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-9edd360f5f7e1614
Observed on the stored rows ?
- Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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: 4, 2; 3: 11, 14, 3; 4: 30, 56, 30, 4; 5: 73, 188, 162, 52, 5; 6: 176, 566, 672, 356, 80, 6; 7: 407, 1586, 2409, 1788, 665, 114, 7; 8: 930, 4236, 7830, 7504, 3950, 1116, 154, 8; 9: 2085, 10896, 23760, 27944, 19130, 7680, 1736, 200, 9;
%F FormulaThe row polynomials satisfy (2*n^2-3*n+1-(n-1)*x)*P(n,x) = -(8*n^2-42*n+40-(10*n-18)*x+2*x^2)*P(n-1,x) + (18*n-27+(20*n-39)*x+(2*n-13)*x^2-x^3)*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[(-(8*n^2 - 42*n + 40 - (10*n - 18)*x + 2*x^2)*P[n - 1, x] + (18*n - 27 + (20*n - 39)*x + (2*n - 13)*x^2 - x^3)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1 - (n - 1)*x)]];
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] = (-(8*n^2 - 42*n + 40 - (10*n - 18)*x + 2*x^2)*v[i - 1] + (18*n - 27 + (20*n - 39)*x + (2*n - 13)*x^2 - x^3)*deriv(v[i - 1], x))/(2*n^2 - 3*n + 1 - (n - 1)*x)); 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 Laguerre’s theorem on differential operators
-
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
-
Laguerre’s theorem on differential operators [SP14] polytool ? step ›
-
from
shift_x_plus_one of A Jacobsthal triangle.?
-
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.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
A Jacobsthal triangle.?
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Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
-
-
from
A Jacobsthal triangle.?
-
-
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 ›
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
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
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
-
from
Pascal-(1,2,1) array read by antidiagonals.?
-
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 ›
-
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.?
-
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)
- ↑ 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.?
-
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) 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)
- ↑ Affine, reversal and Möbius substitutions (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)
- ↑ 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.?
-
Geometric basis map (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)
- ↑ Affine, reversal and Möbius substitutions (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)
- ↑ 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.?
-
Geometric basis map (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)
- ↑ Affine, reversal and Möbius substitutions (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)
- ↑ 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.?
-
Geometric basis map (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)
- ↑ Affine, reversal and Möbius substitutions (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)
- ↑ 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.?
-
-
from
shift_x_plus_one of A Jacobsthal triangle.?
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.