A099097 → basis A013609 → D
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(120 + 216\,x - 72\,n - 108\,n x + 6\,n^{2}\right)\,P_{n}(x) = \left(-572 - 1728\,x - 1296\,x^{2} + 114\,n + 180\,n x - 4\,n^{2}\right)\,P_{n-1}(x) + \left(-143 - 718\,x - 1188\,x^{2} - 648\,x^{3} + 13\,n + 44\,n x + 36\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 6\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 0 | ||||||||||||||
| 2 | 6 | 6 | |||||||||||||
| 3 | 38 | 72 | 110 | ||||||||||||
| 4 | 186 | 696 | 648 | 1530 | |||||||||||
| 5 | 814 | 4544 | 8424 | 5184 | 18966 | ||||||||||
| 6 | 3348 | 24840 | 68904 | 84672 | 38880 | 220644 | |||||||||
| 7 | 13236 | 122496 | 452328 | 832896 | 764640 | 279936 | 2465532 | ||||||||
| 8 | 50910 | 564552 | 2602872 | 6385728 | 8791200 | 6438528 | 1959552 | 26793342 | |||||||
| 9 | 191906 | 2480016 | 13709400 | 42019264 | 77113440 | 84727296 | 51601536 | 13436928 | 285279786 | ||||||
| 10 | 712302 | 10511112 | 67744872 | 249059712 | 571240320 | 836936064 | 764878464 | 398628864 | 90699264 | 2990410974 | |||||
| 11 | 2611774 | 43328600 | 318984072 | 1367715776 | 3763830400 | 6893538048 | 8402486400 | 6572150784 | 2993075712 | 604661760 | 30962383326 | ||||
| 12 | 9482184 | 174687408 | 1446187824 | 7084732608 | 22743249120 | 49988105856 | 76179954816 | 79480424448 | 54328859136 | 21969377280 | 3990767616 | 317395828296 | |||
| 13 | 34144968 | 691624128 | 6359720016 | 35042040384 | 128548361760 | 329642442240 | 602944013952 | 786595940352 | 717259857408 | 435356467200 | 158300448768 | 26121388032 | 3226896449208 | ||
| 14 | 122111574 | 2697228144 | 27274027248 | 166945114752 | 688915468800 | 2019047435136 | 4309148006016 | 6747867095040 | 7694461983744 | 6230501959680 | 3400577427456 | 1123219685376 | 169789022208 | 32580566565174 | |
| 15 | 434149790 | 10385227320 | 114531727440 | 771028856000 | 3534988229760 | 11655492191232 | 28432885776512 | 51956798441472 | 71118589565952 | 72017186365440 | 52439129892864 | 25999488221184 | 7866891362304 | 1097098297344 | 327014928304614 |
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Technical details
- Node id
n-d02d9f98e06db66b· revision 0 · JSON- Definition source
derived:derivative- Fingerprint
f6-f102e7ffe807ea4c
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: 6; 3: 38, 72; 4: 186, 696, 648; 5: 814, 4544, 8424, 5184; 6: 3348, 24840, 68904, 84672, 38880; 7: 13236, 122496, 452328, 832896, 764640, 279936; 8: 50910, 564552, 2602872, 6385728, 8791200, 6438528, 1959552; 9: 191906, 2480016, 13709400, 42019264, 77113440, 84727296, 51601536, 13436928; 10: 712302, 10511112, 67744872, 249059712, 571240320, 836936064, 764878464, 398628864, 90699264;
%F FormulaThe row polynomials satisfy (6*n^2-72*n+120-(108*n-216)*x)*P(n,x) = -(4*n^2-114*n+572-(180*n-1728)*x+1296*x^2)*P(n-1,x) + (13*n-143+(44*n-718)*x+(36*n-1188)*x^2-648*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 6.
%t MathematicaP[1, x_] := 0;
P[2, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 114*n + 572 - (180*n - 1728)*x + 1296*x^2)*P[n - 1, x] + (13*n - 143 + (44*n - 718)*x + (36*n - 1188)*x^2 - 648*x^3)*D[P[n - 1, x], x])/(6*n^2 - 72*n + 120 - (108*n - 216)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 0; v[2] = 6; for(n = 3, N, my(i = n); v[i] = (-(4*n^2 - 114*n + 572 - (180*n - 1728)*x + 1296*x^2)*v[i - 1] + (13*n - 143 + (44*n - 718)*x + (36*n - 1188)*x^2 - 648*x^3)*deriv(v[i - 1], x))/(6*n^2 - 72*n + 120 - (108*n - 216)*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_A013609 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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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)
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- ↑ Liu–Wang theorem for three-term recurrences (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)
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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)
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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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from
basis_map_A013609 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.