A153651 → basis A013609
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-2 + n\right)\,P_{n}(x) = \left(-286 - 572\,x + 143\,n + 286\,n x\right)\,P_{n-2}(x) + \left(-4 - 4\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 26 + 26\,x\)
\(P_{2}(x) = 338 + 676\,x + 8\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 2 | 2 | |||||||
| 1 | 26 | 26 | 52 | ||||||
| 2 | 338 | 676 | 8 | 1022 | |||||
| 3 | 4394 | 13182 | 8804 | 16 | 26396 | ||||
| 4 | 57122 | 228488 | 238452 | 19928 | 32 | 544022 | |||
| 5 | 742586 | 3712930 | 5962904 | 3036992 | 44496 | 64 | 13499972 | ||
| 6 | 9653618 | 57921708 | 118797872 | 89046768 | 11866960 | 98272 | 128 | 287385326 | |
| 7 | 125497034 | 878479238 | 2268032412 | 2555369680 | 1076770096 | 36665472 | 215104 | 256 | 6941029292 |
Compute more
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Technical details
- Node id
n-85cb74c94a93cf11· revision 0 · JSON- Definition source
derived:basis_map_A013609- Fingerprint
f6-c52a40314e6c14a2
Observed on the stored rows ?
- Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
- 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: 2; 1: 26, 26; 2: 338, 676, 8; 3: 4394, 13182, 8804, 16; 4: 57122, 228488, 238452, 19928, 32; 5: 742586, 3712930, 5962904, 3036992, 44496, 64; 6: 9653618, 57921708, 118797872, 89046768, 11866960, 98272, 128; 7: 125497034, 878479238, 2268032412, 2555369680, 1076770096, 36665472, 215104, 256; 8: 1631461442, 13051691536, 39846727484, 56356683400, 34428630480, 5634874592, 101885248, 467328, 512; 9: 21208998746, 190880988714, 681370535024, 1206481955840, 1066384479968, 393326420096, 21990604544, 266261504, 1008896, 1024;
%F FormulaThe row polynomials satisfy (n-2)*P(n,x) = (143*n-286+(286*n-572)*x)*P(n-2,x) + (2*n-4+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 26 + 26*x, P(2,x) = 338 + 676*x + 8*x^2.
%t MathematicaP[0, x_] := 2;
P[1, x_] := 26 + 26*x;
P[2, x_] := 338 + 676*x + 8*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((143*n - 286 + (286*n - 572)*x)*P[n - 2, x] + (2*n - 4 + (2*n - 4)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 26 + 26*x; v[3] = 338 + 676*x + 8*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((143*n - 286 + (286*n - 572)*x)*v[i - 2] + (2*n - 4 + (2*n - 4)*x)*v[i - 1])/(n - 2)); 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 Geometric basis map (elementary)
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
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Geometric basis map (elementary) polytool ? step ›
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from
Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)*prime(j)*T(n-2, k-1) with j=6, read by rows.?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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from
Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)*prime(j)*T(n-2, k-1) with j=6, read by rows.?
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