A257625 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(1 - 4\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-9 - 540\,x - 135\,x^{2} + 360\,n x + 144\,n x^{2} + 36\,n^{2} + 432\,n^{2} x + 1368\,n^{2} x^{2} - 288\,n^{3} x - 2880\,n^{3} x^{2} + 1296\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(144\,x - 540\,x^{2} - 180\,x^{3} + 72\,n x^{2} + 72\,n x^{3} - 144\,n^{2} x + 432\,n^{2} x^{2} + 2016\,n^{2} x^{3} + 288\,n^{3} x^{2} - 3168\,n^{3} x^{3} + 576\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(108\,x^{2} - 72\,x^{3} - 36\,x^{4} - 144\,n^{2} x^{2} - 288\,n^{2} x^{3} + 432\,n^{2} x^{4} + 576\,n^{3} x^{3} - 576\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(6 - 12\,x - 12\,n + 72\,n x - 144\,n^{2} x + 96\,n^{3} x\right)\,P_{n-1}(x) + \left(-12\,x - 12\,x^{2} - 24\,n x + 24\,n x^{2} + 48\,n^{2} x + 48\,n^{2} x^{2} - 96\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 12\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 3 | 12 | 15 | ||||||
| 2 | 9 | 216 | 81 | 306 | |||||
| 3 | 27 | 2484 | 5589 | 432 | 8532 | ||||
| 4 | 81 | 24624 | 167670 | 98496 | 2025 | 292896 | |||
| 5 | 243 | 230364 | 3678534 | 6539616 | 1439775 | 8748 | 11897280 | ||
| 6 | 729 | 2105352 | 69172623 | 274663872 | 192146175 | 18948168 | 35721 | 557072640 | |
| 7 | 2187 | 19061892 | 1193911731 | 9088227216 | 14200355025 | 4775646924 | 233508177 | 139968 | 29510853120 |
Compute more
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Technical details
- Node id
n-bfe4e5560512fd9d· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-5d3f47477444bdf9
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: 3, 12; 2: 9, 216, 81; 3: 27, 2484, 5589, 432; 4: 81, 24624, 167670, 98496, 2025; 5: 243, 230364, 3678534, 6539616, 1439775, 8748; 6: 729, 2105352, 69172623, 274663872, 192146175, 18948168, 35721; 7: 2187, 19061892, 1193911731, 9088227216, 14200355025, 4775646924, 233508177, 139968; 8: 6561, 171950688, 19581356988, 260895593088, 766819171350, 587015084448, 106609610268, 2751211008, 531441; 9: 19683, 1548894636, 311130361980, 6836448206016, 34156546645050, 49185427168872, 20936622630924, 2212482574080, 31365116379, 1968300;
%F FormulaThe row polynomials satisfy (4*n^2-4*n+1)*P(n,x) = (36*n^2-9-(288*n^3-432*n^2-360*n+540)*x+(1296*n^4-2880*n^3+1368*n^2+144*n-135)*x^2)*P(n-2,x) - ((144*n^2-144)*x-(288*n^3+432*n^2+72*n-540)*x^2-(576*n^4-3168*n^3+2016*n^2+72*n-180)*x^3)*d/dx P(n-2,x) - ((144*n^2-108)*x^2-(576*n^3-288*n^2-72)*x^3+(576*n^3-432*n^2+36)*x^4)*d^2/dx^2 P(n-2,x) - (12*n-6-(96*n^3-144*n^2+72*n-12)*x)*P(n-1,x) + ((48*n^2-24*n-12)*x-(96*n^3-48*n^2-24*n+12)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 3 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[((36*n^2 - 9 - (288*n^3 - 432*n^2 - 360*n + 540)*x + (1296*n^4 - 2880*n^3 + 1368*n^2 + 144*n - 135)*x^2)*P[n - 2, x] - ((144*n^2 - 144)*x - (288*n^3 + 432*n^2 + 72*n - 540)*x^2 - (576*n^4 - 3168*n^3 + 2016*n^2 + 72*n - 180)*x^3)*D[P[n - 2, x], x] - ((144*n^2 - 108)*x^2 - (576*n^3 - 288*n^2 - 72)*x^3 + (576*n^3 - 432*n^2 + 36)*x^4)*D[P[n - 2, x], {x, 2}] - (12*n - 6 - (96*n^3 - 144*n^2 + 72*n - 12)*x)*P[n - 1, x] + ((48*n^2 - 24*n - 12)*x - (96*n^3 - 48*n^2 - 24*n + 12)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 4*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = ((36*n^2 - 9 - (288*n^3 - 432*n^2 - 360*n + 540)*x + (1296*n^4 - 2880*n^3 + 1368*n^2 + 144*n - 135)*x^2)*v[i - 2] - ((144*n^2 - 144)*x - (288*n^3 + 432*n^2 + 72*n - 540)*x^2 - (576*n^4 - 3168*n^3 + 2016*n^2 + 72*n - 180)*x^3)*deriv(v[i - 2], x) - ((144*n^2 - 108)*x^2 - (576*n^3 - 288*n^2 - 72)*x^3 + (576*n^3 - 432*n^2 + 36)*x^4)*deriv(deriv(v[i - 2], x), x) - (12*n - 6 - (96*n^3 - 144*n^2 + 72*n - 12)*x)*v[i - 1] + ((48*n^2 - 24*n - 12)*x - (96*n^3 - 48*n^2 - 24*n + 12)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 4*n + 1)); 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 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
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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.