A154929 → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(4 + 2\,x - n - 4\,n^{2} - 2\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(20 + 4\,x - 58\,n - 36\,n x - 6\,n x^{2} + 22\,n^{2} + 8\,n^{2} x - 2\,n^{3}\right)\,P_{n-1}(x) + \left(10 + 12\,x + 2\,x^{2} - 25\,n - 40\,n x - 17\,n x^{2} - 2\,n x^{3} + 5\,n^{2} + 6\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
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 | 10 | 12 | 3 | 25 | |||||||||||
| 4 | 22 | 42 | 24 | 4 | 92 | ||||||||||
| 5 | 45 | 118 | 108 | 40 | 5 | 316 | |||||||||
| 6 | 88 | 294 | 372 | 220 | 60 | 6 | 1040 | ||||||||
| 7 | 167 | 678 | 1098 | 900 | 390 | 84 | 7 | 3324 | |||||||
| 8 | 310 | 1482 | 2928 | 3080 | 1850 | 630 | 112 | 8 | 10400 | ||||||
| 9 | 566 | 3114 | 7266 | 9348 | 7215 | 3402 | 952 | 144 | 9 | 32016 | |||||
| 10 | 1020 | 6348 | 17088 | 26020 | 24600 | 14910 | 5768 | 1368 | 180 | 10 | 97312 | ||||
| 11 | 1819 | 12634 | 38547 | 67840 | 76025 | 56490 | 28084 | 9192 | 1890 | 220 | 11 | 292752 | |||
| 12 | 3216 | 24660 | 84108 | 168040 | 217950 | 192150 | 117152 | 49248 | 13950 | 2530 | 264 | 12 | 873280 | ||
| 13 | 5645 | 47364 | 178596 | 399420 | 588825 | 601410 | 435316 | 224424 | 81585 | 20350 | 3300 | 312 | 13 | 2586560 | |
| 14 | 9848 | 89746 | 370740 | 917732 | 1516100 | 1761690 | 1478288 | 905064 | 403416 | 129030 | 28732 | 4212 | 364 | 14 | 7614976 |
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Technical details
- Node id
n-c67e979e07e2ce8f· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-69c92bdc559cc6fe
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: 10, 12, 3; 4: 22, 42, 24, 4; 5: 45, 118, 108, 40, 5; 6: 88, 294, 372, 220, 60, 6; 7: 167, 678, 1098, 900, 390, 84, 7; 8: 310, 1482, 2928, 3080, 1850, 630, 112, 8; 9: 566, 3114, 7266, 9348, 7215, 3402, 952, 144, 9;
%F FormulaThe row polynomials satisfy (n^3-4*n^2-n+4-(2*n^2-2)*x)*P(n,x) = -(2*n^3-22*n^2+58*n-20-(8*n^2-36*n+4)*x+6*n*x^2)*P(n-1,x) + (5*n^2-25*n+10+(6*n^2-40*n+12)*x+(n^2-17*n+2)*x^2-2*n*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[(-(2*n^3 - 22*n^2 + 58*n - 20 - (8*n^2 - 36*n + 4)*x + 6*n*x^2)*P[n - 1, x] + (5*n^2 - 25*n + 10 + (6*n^2 - 40*n + 12)*x + (n^2 - 17*n + 2)*x^2 - 2*n*x^3)*D[P[n - 1, x], x])/(n^3 - 4*n^2 - n + 4 - (2*n^2 - 2)*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] = (-(2*n^3 - 22*n^2 + 58*n - 20 - (8*n^2 - 36*n + 4)*x + 6*n*x^2)*v[i - 1] + (5*n^2 - 25*n + 10 + (6*n^2 - 40*n + 12)*x + (n^2 - 17*n + 2)*x^2 - 2*n*x^3)*deriv(v[i - 1], x))/(n^3 - 4*n^2 - n + 4 - (2*n^2 - 2)*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]
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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
shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Geometric basis map (elementary) polytool ? step ›
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from
basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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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
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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from
basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).?
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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
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
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
delannoy?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
delannoy?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
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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
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
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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
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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)
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from
delannoy?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
-
-
from
delannoy?
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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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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
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).?
-
Geometric basis map (elementary) polytool ? step ›
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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).?
-
Geometric basis map (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).?
-
Geometric basis map (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
shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(k,n-k).?
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.