A142458 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(4 - 12\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(-4 - 400\,x - 60\,x^{2} - 6\,n + 180\,n x - 324\,n x^{2} + 18\,n^{2} + 216\,n^{2} x + 1701\,n^{2} x^{2} - 108\,n^{3} x - 2106\,n^{3} x^{2} + 729\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(84\,x - 380\,x^{2} - 68\,x^{3} + 18\,n x - 27\,n x^{2} - 492\,n x^{3} - 54\,n^{2} x + 216\,n^{2} x^{2} + 2079\,n^{2} x^{3} + 108\,n^{3} x^{2} - 1998\,n^{3} x^{3} + 324\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(72\,x^{2} - 60\,x^{3} - 12\,x^{4} + 27\,n x^{2} + 81\,n x^{3} - 108\,n x^{4} - 81\,n^{2} x^{2} - 324\,n^{2} x^{3} + 405\,n^{2} x^{4} + 324\,n^{3} x^{3} - 324\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(8 - 32\,x - 6\,n + 144\,n x - 9\,n^{2} - 216\,n^{2} x + 108\,n^{3} x\right)\,P_{n-1}(x) + \left(-12\,x - 16\,x^{2} - 45\,n x + 54\,n^{2} x + 108\,n^{2} x^{2} - 108\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 4 | 5 | ||||||
| 2 | 1 | 32 | 9 | 42 | |||||
| 3 | 1 | 156 | 351 | 16 | 524 | ||||
| 4 | 1 | 664 | 4914 | 2656 | 25 | 8260 | |||
| 5 | 1 | 2708 | 49338 | 87712 | 16925 | 36 | 156720 | ||
| 6 | 1 | 10896 | 424575 | 1754240 | 1179375 | 98064 | 49 | 3467200 | |
| 7 | 1 | 43660 | 3364281 | 27354800 | 42741875 | 13457124 | 534835 | 64 | 87496640 |
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Technical details
- Node id
n-841d3511dcce72f4· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-34bff62b9cc31312
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: 1; 1: 1, 4; 2: 1, 32, 9; 3: 1, 156, 351, 16; 4: 1, 664, 4914, 2656, 25; 5: 1, 2708, 49338, 87712, 16925, 36; 6: 1, 10896, 424575, 1754240, 1179375, 98064, 49; 7: 1, 43660, 3364281, 27354800, 42741875, 13457124, 534835, 64; 8: 1, 174728, 25416432, 369243104, 1111288750, 830796984, 138378352, 2795648, 81; 9: 1, 699012, 186564060, 4550941632, 23677831350, 34096077144, 13937258748, 1326677760, 14154993, 100;
%F FormulaThe row polynomials satisfy (9*n^2-12*n+4)*P(n,x) = (18*n^2-6*n-4-(108*n^3-216*n^2-180*n+400)*x+(729*n^4-2106*n^3+1701*n^2-324*n-60)*x^2)*P(n-2,x) - ((54*n^2-18*n-84)*x-(108*n^3+216*n^2-27*n-380)*x^2-(324*n^4-1998*n^3+2079*n^2-492*n-68)*x^3)*d/dx P(n-2,x) - ((81*n^2-27*n-72)*x^2-(324*n^3-324*n^2+81*n-60)*x^3+(324*n^3-405*n^2+108*n+12)*x^4)*d^2/dx^2 P(n-2,x) - (9*n^2+6*n-8-(108*n^3-216*n^2+144*n-32)*x)*P(n-1,x) + ((54*n^2-45*n-12)*x-(108*n^3-108*n^2+16)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n^2 - 6*n - 4 - (108*n^3 - 216*n^2 - 180*n + 400)*x + (729*n^4 - 2106*n^3 + 1701*n^2 - 324*n - 60)*x^2)*P[n - 2, x] - ((54*n^2 - 18*n - 84)*x - (108*n^3 + 216*n^2 - 27*n - 380)*x^2 - (324*n^4 - 1998*n^3 + 2079*n^2 - 492*n - 68)*x^3)*D[P[n - 2, x], x] - ((81*n^2 - 27*n - 72)*x^2 - (324*n^3 - 324*n^2 + 81*n - 60)*x^3 + (324*n^3 - 405*n^2 + 108*n + 12)*x^4)*D[P[n - 2, x], {x, 2}] - (9*n^2 + 6*n - 8 - (108*n^3 - 216*n^2 + 144*n - 32)*x)*P[n - 1, x] + ((54*n^2 - 45*n - 12)*x - (108*n^3 - 108*n^2 + 16)*x^2)*D[P[n - 1, x], x])/(9*n^2 - 12*n + 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((18*n^2 - 6*n - 4 - (108*n^3 - 216*n^2 - 180*n + 400)*x + (729*n^4 - 2106*n^3 + 1701*n^2 - 324*n - 60)*x^2)*v[i - 2] - ((54*n^2 - 18*n - 84)*x - (108*n^3 + 216*n^2 - 27*n - 380)*x^2 - (324*n^4 - 1998*n^3 + 2079*n^2 - 492*n - 68)*x^3)*deriv(v[i - 2], x) - ((81*n^2 - 27*n - 72)*x^2 - (324*n^3 - 324*n^2 + 81*n - 60)*x^3 + (324*n^3 - 405*n^2 + 108*n + 12)*x^4)*deriv(deriv(v[i - 2], x), x) - (9*n^2 + 6*n - 8 - (108*n^3 - 216*n^2 + 144*n - 32)*x)*v[i - 1] + ((54*n^2 - 45*n - 12)*x - (108*n^3 - 108*n^2 + 16)*x^2)*deriv(v[i - 1], x))/(9*n^2 - 12*n + 4)); 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]
-
Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, 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 ≥ 1
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.