A142459 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(49 - 56\,n + 16\,n^{2}\right)\,P_{n}(x) = \left(63 - 1188\,x + 18865\,x^{2} - 120\,n - 1328\,n x - 47824\,n x^{2} + 48\,n^{2} + 1344\,n^{2} x + 43232\,n^{2} x^{2} - 256\,n^{3} x - 16640\,n^{3} x^{2} + 2304\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(72\,x - 652\,x^{2} + 18620\,x^{3} + 320\,n x - 848\,n x^{2} - 43232\,n x^{3} - 128\,n^{2} x - 64\,n^{2} x^{2} + 34304\,n^{2} x^{3} + 256\,n^{3} x^{2} - 10752\,n^{3} x^{3} + 1024\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-144\,x^{2} - 2992\,x^{3} + 3136\,x^{4} + 640\,n x^{2} + 6080\,n x^{3} - 6720\,n x^{4} - 256\,n^{2} x^{2} - 4352\,n^{2} x^{3} + 4608\,n^{2} x^{4} + 1024\,n^{3} x^{3} - 1024\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-14 - 1372\,x + 64\,n + 2352\,n x - 32\,n^{2} - 1344\,n^{2} x + 256\,n^{3} x\right)\,P_{n-1}(x) + \left(232\,x + 588\,x^{2} - 384\,n x - 1456\,n x^{2} + 128\,n^{2} x + 1088\,n^{2} x^{2} - 256\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 4 | 5 | ||||||
| 3 | 1 | 40 | 9 | 50 | |||||
| 4 | 1 | 236 | 531 | 16 | 784 | ||||
| 5 | 1 | 1232 | 9558 | 4928 | 25 | 15744 | |||
| 6 | 1 | 6228 | 122058 | 216992 | 38925 | 36 | 384240 | ||
| 7 | 1 | 31224 | 1336743 | 5641792 | 3713175 | 281016 | 49 | 11004000 | |
| 8 | 1 | 156220 | 13506021 | 113742640 | 177722875 | 54024084 | 1913695 | 64 | 361065600 |
Compute more
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Technical details
- Node id
n-2793b6a4aab90b67· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-fffff3ff3ecdc6fc
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 -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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 1, 4; 3: 1, 40, 9; 4: 1, 236, 531, 16; 5: 1, 1232, 9558, 4928, 25; 6: 1, 6228, 122058, 216992, 38925, 36; 7: 1, 31224, 1336743, 5641792, 3713175, 281016, 49; 8: 1, 156220, 13506021, 113742640, 177722875, 54024084, 1913695, 64; 9: 1, 781216, 130341564, 1982879104, 6042577750, 4461477984, 709637404, 12499456, 81; 10: 1, 3906212, 1224048420, 31570386752, 167787042350, 241613340984, 96684309428, 8704344320, 79100793, 100;
%F FormulaThe row polynomials satisfy (16*n^2-56*n+49)*P(n,x) = (48*n^2-120*n+63-(256*n^3-1344*n^2+1328*n+1188)*x+(2304*n^4-16640*n^3+43232*n^2-47824*n+18865)*x^2)*P(n-2,x) - ((128*n^2-320*n-72)*x-(256*n^3-64*n^2-848*n-652)*x^2-(1024*n^4-10752*n^3+34304*n^2-43232*n+18620)*x^3)*d/dx P(n-2,x) - ((256*n^2-640*n+144)*x^2-(1024*n^3-4352*n^2+6080*n-2992)*x^3+(1024*n^3-4608*n^2+6720*n-3136)*x^4)*d^2/dx^2 P(n-2,x) - (32*n^2-64*n+14-(256*n^3-1344*n^2+2352*n-1372)*x)*P(n-1,x) + ((128*n^2-384*n+232)*x-(256*n^3-1088*n^2+1456*n-588)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((48*n^2 - 120*n + 63 - (256*n^3 - 1344*n^2 + 1328*n + 1188)*x + (2304*n^4 - 16640*n^3 + 43232*n^2 - 47824*n + 18865)*x^2)*P[n - 2, x] - ((128*n^2 - 320*n - 72)*x - (256*n^3 - 64*n^2 - 848*n - 652)*x^2 - (1024*n^4 - 10752*n^3 + 34304*n^2 - 43232*n + 18620)*x^3)*D[P[n - 2, x], x] - ((256*n^2 - 640*n + 144)*x^2 - (1024*n^3 - 4352*n^2 + 6080*n - 2992)*x^3 + (1024*n^3 - 4608*n^2 + 6720*n - 3136)*x^4)*D[P[n - 2, x], {x, 2}] - (32*n^2 - 64*n + 14 - (256*n^3 - 1344*n^2 + 2352*n - 1372)*x)*P[n - 1, x] + ((128*n^2 - 384*n + 232)*x - (256*n^3 - 1088*n^2 + 1456*n - 588)*x^2)*D[P[n - 1, x], x])/(16*n^2 - 56*n + 49)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = ((48*n^2 - 120*n + 63 - (256*n^3 - 1344*n^2 + 1328*n + 1188)*x + (2304*n^4 - 16640*n^3 + 43232*n^2 - 47824*n + 18865)*x^2)*v[i - 2] - ((128*n^2 - 320*n - 72)*x - (256*n^3 - 64*n^2 - 848*n - 652)*x^2 - (1024*n^4 - 10752*n^3 + 34304*n^2 - 43232*n + 18620)*x^3)*deriv(v[i - 2], x) - ((256*n^2 - 640*n + 144)*x^2 - (1024*n^3 - 4352*n^2 + 6080*n - 2992)*x^3 + (1024*n^3 - 4608*n^2 + 6720*n - 3136)*x^4)*deriv(deriv(v[i - 2], x), x) - (32*n^2 - 64*n + 14 - (256*n^3 - 1344*n^2 + 2352*n - 1372)*x)*v[i - 1] + ((128*n^2 - 384*n + 232)*x - (256*n^3 - 1088*n^2 + 1456*n - 588)*x^2)*deriv(v[i - 1], x))/(16*n^2 - 56*n + 49)); 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]
-
Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
Proof chains ? JSON
Related ?
Same fingerprint ?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [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
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.