A099097 → basis A007318 → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(60 + 54\,x - 36\,n - 27\,n x + 3\,n^{2}\right)\,P_{n}(x) = \left(-286 - 432\,x - 162\,x^{2} + 57\,n + 45\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-143 - 359\,x - 297\,x^{2} - 81\,x^{3} + 13\,n + 22\,n x + 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 3\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 0 | ||||||||||||||
| 2 | 3 | 3 | |||||||||||||
| 3 | 19 | 18 | 37 | ||||||||||||
| 4 | 93 | 174 | 81 | 348 | |||||||||||
| 5 | 407 | 1136 | 1053 | 324 | 2920 | ||||||||||
| 6 | 1674 | 6210 | 8613 | 5292 | 1215 | 23004 | |||||||||
| 7 | 6618 | 30624 | 56541 | 52056 | 23895 | 4374 | 174108 | ||||||||
| 8 | 25455 | 141138 | 325359 | 399108 | 274725 | 100602 | 15309 | 1281696 | |||||||
| 9 | 95953 | 620004 | 1713675 | 2626204 | 2409795 | 1323864 | 403137 | 52488 | 9245120 | ||||||
| 10 | 356151 | 2627778 | 8468109 | 15566232 | 17851260 | 13077126 | 5975613 | 1557144 | 177147 | 65656560 | |||||
| 11 | 1305887 | 10832150 | 39873009 | 85482236 | 117619700 | 107711532 | 65644425 | 25672464 | 5845851 | 590490 | 460577744 | ||||
| 12 | 4741092 | 43671852 | 180773478 | 442795788 | 710726535 | 781064154 | 595155897 | 310470408 | 106111053 | 21454470 | 1948617 | 3198913344 | |||
| 13 | 17072484 | 172906032 | 794965002 | 2190127524 | 4017136305 | 5150663160 | 4710500109 | 3072640392 | 1400898159 | 425152800 | 77295141 | 6377292 | 22035734400 | ||
| 14 | 61055787 | 674307036 | 3409253406 | 10434069672 | 21528608400 | 31547616174 | 33665218797 | 26358855840 | 15028246062 | 6084474570 | 1660438197 | 274223556 | 20726199 | 150747093696 | |
| 15 | 217074895 | 2596306830 | 14316465930 | 48189303500 | 110468382180 | 182117065488 | 222131920129 | 202956243912 | 138903495246 | 70329283560 | 25605043893 | 6347531304 | 960313887 | 66961566 | 1025205392320 |
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Technical details
- Node id
n-8c7cdc667bc71c38· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-3246cf40c1954eb1
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 -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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 0; 2: 3; 3: 19, 18; 4: 93, 174, 81; 5: 407, 1136, 1053, 324; 6: 1674, 6210, 8613, 5292, 1215; 7: 6618, 30624, 56541, 52056, 23895, 4374; 8: 25455, 141138, 325359, 399108, 274725, 100602, 15309; 9: 95953, 620004, 1713675, 2626204, 2409795, 1323864, 403137, 52488; 10: 356151, 2627778, 8468109, 15566232, 17851260, 13077126, 5975613, 1557144, 177147;
%F FormulaThe row polynomials satisfy (3*n^2-36*n+60-(27*n-54)*x)*P(n,x) = -(2*n^2-57*n+286-(45*n-432)*x+162*x^2)*P(n-1,x) + (13*n-143+(22*n-359)*x+(9*n-297)*x^2-81*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.
%t MathematicaP[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 57*n + 286 - (45*n - 432)*x + 162*x^2)*P[n - 1, x] + (13*n - 143 + (22*n - 359)*x + (9*n - 297)*x^2 - 81*x^3)*D[P[n - 1, x], x])/(3*n^2 - 36*n + 60 - (27*n - 54)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 57*n + 286 - (45*n - 432)*x + 162*x^2)*v[i - 1] + (13*n - 143 + (22*n - 359)*x + (9*n - 297)*x^2 - 81*x^3)*deriv(v[i - 1], x))/(3*n^2 - 36*n + 60 - (27*n - 54)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
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Coefficients of one sign?
Every nonzero row, n ≥ 1: 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 ≥ 1, 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_A007318 of Riordan array (1, 3+x).?
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Geometric basis map (elementary) polytool ? step ›
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from
Riordan array (1, 3+x).?
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Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
no recorded property covers this requirement
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from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
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Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
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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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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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from
Riordan array (1, 3+x).?
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Geometric basis map (elementary) polytool ? step ›
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from
Riordan array (1, 3+x).?
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (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)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
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from
Riordan array (1, 3+x).?
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Geometric basis map (elementary) polytool ? step ›
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from
Riordan array (1, 3+x).?
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (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)
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- ↑ 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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from
Riordan array (1, 3+x).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Riordan array (1, 3+x).?
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (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)
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- ↑ Geometric basis map (elementary) (above)
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- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
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from
Riordan array (1, 3+x).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Riordan array (1, 3+x).?
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (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)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
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from
Riordan array (1, 3+x).?
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from
basis_map_A007318 of Riordan array (1, 3+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.