A257614 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(9 - 30\,n + 25\,n^{2}\right)\,P_{n}(x) = \left(-36 - 2784\,x - 576\,x^{2} - 30\,n + 1420\,n x - 960\,n x^{2} + 150\,n^{2} + 1800\,n^{2} x + 10025\,n^{2} x^{2} - 1000\,n^{3} x - 14750\,n^{3} x^{2} + 5625\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(645\,x - 2706\,x^{2} - 711\,x^{3} + 100\,n x + 5\,n x^{2} - 1860\,n x^{3} - 500\,n^{2} x + 1700\,n^{2} x^{2} + 13025\,n^{2} x^{3} + 1000\,n^{3} x^{2} - 14750\,n^{3} x^{3} + 2500\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(525\,x^{2} - 390\,x^{3} - 135\,x^{4} + 125\,n x^{2} + 325\,n x^{3} - 450\,n x^{4} - 625\,n^{2} x^{2} - 2000\,n^{2} x^{3} + 2625\,n^{2} x^{4} + 2500\,n^{3} x^{3} - 2500\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(36 - 108\,x - 45\,n + 540\,n x - 25\,n^{2} - 900\,n^{2} x + 500\,n^{3} x\right)\,P_{n-1}(x) + \left(-60\,x - 72\,x^{2} - 175\,n x + 60\,n x^{2} + 250\,n^{2} x + 400\,n^{2} x^{2} - 500\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 8 | 10 | ||||||
| 2 | 4 | 112 | 36 | 152 | |||||
| 3 | 8 | 976 | 2196 | 128 | 3308 | ||||
| 4 | 16 | 7376 | 52704 | 29504 | 400 | 90000 | |||
| 5 | 32 | 53040 | 914580 | 1625920 | 331500 | 1152 | 2926224 | ||
| 6 | 64 | 374736 | 13600440 | 55281280 | 37779000 | 3372624 | 3136 | 110411280 | |
| 7 | 128 | 2631344 | 185970492 | 1471710080 | 2299547000 | 743881968 | 32233964 | 8192 | 4735983168 |
Compute more
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Technical details
- Node id
n-6f20c4def6bc959e· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-957858e7426cd59f
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: 2, 8; 2: 4, 112, 36; 3: 8, 976, 2196, 128; 4: 16, 7376, 52704, 29504, 400; 5: 32, 53040, 914580, 1625920, 331500, 1152; 6: 64, 374736, 13600440, 55281280, 37779000, 3372624, 3136; 7: 128, 2631344, 185970492, 1471710080, 2299547000, 743881968, 32233964, 8192; 8: 256, 18438352, 2421102672, 33945654976, 101180068000, 76377723696, 13181558992, 295013632, 20736; 9: 512, 129111472, 30588224868, 714809975488, 3658043815300, 5267583094032, 2189105549932, 217516265728, 2614507308, 51200;
%F FormulaThe row polynomials satisfy (25*n^2-30*n+9)*P(n,x) = (150*n^2-30*n-36-(1000*n^3-1800*n^2-1420*n+2784)*x+(5625*n^4-14750*n^3+10025*n^2-960*n-576)*x^2)*P(n-2,x) - ((500*n^2-100*n-645)*x-(1000*n^3+1700*n^2+5*n-2706)*x^2-(2500*n^4-14750*n^3+13025*n^2-1860*n-711)*x^3)*d/dx P(n-2,x) - ((625*n^2-125*n-525)*x^2-(2500*n^3-2000*n^2+325*n-390)*x^3+(2500*n^3-2625*n^2+450*n+135)*x^4)*d^2/dx^2 P(n-2,x) - (25*n^2+45*n-36-(500*n^3-900*n^2+540*n-108)*x)*P(n-1,x) + ((250*n^2-175*n-60)*x-(500*n^3-400*n^2-60*n+72)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((150*n^2 - 30*n - 36 - (1000*n^3 - 1800*n^2 - 1420*n + 2784)*x + (5625*n^4 - 14750*n^3 + 10025*n^2 - 960*n - 576)*x^2)*P[n - 2, x] - ((500*n^2 - 100*n - 645)*x - (1000*n^3 + 1700*n^2 + 5*n - 2706)*x^2 - (2500*n^4 - 14750*n^3 + 13025*n^2 - 1860*n - 711)*x^3)*D[P[n - 2, x], x] - ((625*n^2 - 125*n - 525)*x^2 - (2500*n^3 - 2000*n^2 + 325*n - 390)*x^3 + (2500*n^3 - 2625*n^2 + 450*n + 135)*x^4)*D[P[n - 2, x], {x, 2}] - (25*n^2 + 45*n - 36 - (500*n^3 - 900*n^2 + 540*n - 108)*x)*P[n - 1, x] + ((250*n^2 - 175*n - 60)*x - (500*n^3 - 400*n^2 - 60*n + 72)*x^2)*D[P[n - 1, x], x])/(25*n^2 - 30*n + 9)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((150*n^2 - 30*n - 36 - (1000*n^3 - 1800*n^2 - 1420*n + 2784)*x + (5625*n^4 - 14750*n^3 + 10025*n^2 - 960*n - 576)*x^2)*v[i - 2] - ((500*n^2 - 100*n - 645)*x - (1000*n^3 + 1700*n^2 + 5*n - 2706)*x^2 - (2500*n^4 - 14750*n^3 + 13025*n^2 - 1860*n - 711)*x^3)*deriv(v[i - 2], x) - ((625*n^2 - 125*n - 525)*x^2 - (2500*n^3 - 2000*n^2 + 325*n - 390)*x^3 + (2500*n^3 - 2625*n^2 + 450*n + 135)*x^4)*deriv(deriv(v[i - 2], x), x) - (25*n^2 + 45*n - 36 - (500*n^3 - 900*n^2 + 540*n - 108)*x)*v[i - 1] + ((250*n^2 - 175*n - 60)*x - (500*n^3 - 400*n^2 - 60*n + 72)*x^2)*deriv(v[i - 1], x))/(25*n^2 - 30*n + 9)); 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
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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 ›
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.