A257621 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(1 - 8\,n + 16\,n^{2}\right)\,P_{n}(x) = \left(-9 - 500\,x - 95\,x^{2} + 24\,n + 880\,n x + 656\,n x^{2} + 48\,n^{2} + 576\,n^{2} x - 544\,n^{2} x^{2} - 768\,n^{3} x - 2816\,n^{3} x^{2} + 2304\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(184\,x - 540\,x^{2} - 148\,x^{3} - 192\,n x + 528\,n x^{2} + 928\,n x^{3} - 384\,n^{2} x + 1216\,n^{2} x^{2} - 256\,n^{2} x^{3} + 768\,n^{3} x^{2} - 4608\,n^{3} x^{3} + 1024\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(112\,x^{2} - 80\,x^{3} - 32\,x^{4} - 128\,n x^{2} - 64\,n x^{3} + 192\,n x^{4} - 256\,n^{2} x^{2} + 256\,n^{2} x^{3} + 1024\,n^{3} x^{3} - 1024\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(6 - 4\,x - 32\,n + 48\,n x + 32\,n^{2} - 192\,n^{2} x + 256\,n^{3} x\right)\,P_{n-1}(x) + \left(-24\,x - 12\,x^{2} + 80\,n x^{2} + 128\,n^{2} x - 64\,n^{2} x^{2} - 256\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 12\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 3 | 12 | 15 | ||||||
| 2 | 9 | 168 | 81 | 258 | |||||
| 3 | 27 | 1572 | 3537 | 432 | 5568 | ||||
| 4 | 81 | 12624 | 77814 | 50496 | 2025 | 143040 | |||
| 5 | 243 | 94524 | 1282014 | 2279136 | 590775 | 8748 | 4255440 | ||
| 6 | 729 | 684024 | 18143055 | 68374080 | 50397375 | 6156216 | 35721 | 143791200 | |
| 7 | 2187 | 4866900 | 234971847 | 1638443280 | 2560067625 | 939887388 | 59619525 | 139968 | 5437998720 |
Compute more
?
Technical details
- Node id
n-6be60b87a855006c· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-7fcfcfdc25807408
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, 168, 81; 3: 27, 1572, 3537, 432; 4: 81, 12624, 77814, 50496, 2025; 5: 243, 94524, 1282014, 2279136, 590775, 8748; 6: 729, 684024, 18143055, 68374080, 50397375, 6156216, 35721; 7: 2187, 4866900, 234971847, 1638443280, 2560067625, 939887388, 59619525, 139968; 8: 6561, 34339488, 2880354492, 34184386944, 97282569750, 76914870624, 15681930012, 549431808, 531441; 9: 19683, 241294956, 34079078700, 651022819776, 3076870231050, 4430693132712, 1993757385564, 242340115200, 4886222859, 1968300;
%F FormulaThe row polynomials satisfy (16*n^2-8*n+1)*P(n,x) = (48*n^2+24*n-9-(768*n^3-576*n^2-880*n+500)*x+(2304*n^4-2816*n^3-544*n^2+656*n-95)*x^2)*P(n-2,x) - ((384*n^2+192*n-184)*x-(768*n^3+1216*n^2+528*n-540)*x^2-(1024*n^4-4608*n^3-256*n^2+928*n-148)*x^3)*d/dx P(n-2,x) - ((256*n^2+128*n-112)*x^2-(1024*n^3+256*n^2-64*n-80)*x^3+(1024*n^3-192*n+32)*x^4)*d^2/dx^2 P(n-2,x) + (32*n^2-32*n+6+(256*n^3-192*n^2+48*n-4)*x)*P(n-1,x) + ((128*n^2-24)*x-(256*n^3+64*n^2-80*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[((48*n^2 + 24*n - 9 - (768*n^3 - 576*n^2 - 880*n + 500)*x + (2304*n^4 - 2816*n^3 - 544*n^2 + 656*n - 95)*x^2)*P[n - 2, x] - ((384*n^2 + 192*n - 184)*x - (768*n^3 + 1216*n^2 + 528*n - 540)*x^2 - (1024*n^4 - 4608*n^3 - 256*n^2 + 928*n - 148)*x^3)*D[P[n - 2, x], x] - ((256*n^2 + 128*n - 112)*x^2 - (1024*n^3 + 256*n^2 - 64*n - 80)*x^3 + (1024*n^3 - 192*n + 32)*x^4)*D[P[n - 2, x], {x, 2}] + (32*n^2 - 32*n + 6 + (256*n^3 - 192*n^2 + 48*n - 4)*x)*P[n - 1, x] + ((128*n^2 - 24)*x - (256*n^3 + 64*n^2 - 80*n + 12)*x^2)*D[P[n - 1, x], x])/(16*n^2 - 8*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] = ((48*n^2 + 24*n - 9 - (768*n^3 - 576*n^2 - 880*n + 500)*x + (2304*n^4 - 2816*n^3 - 544*n^2 + 656*n - 95)*x^2)*v[i - 2] - ((384*n^2 + 192*n - 184)*x - (768*n^3 + 1216*n^2 + 528*n - 540)*x^2 - (1024*n^4 - 4608*n^3 - 256*n^2 + 928*n - 148)*x^3)*deriv(v[i - 2], x) - ((256*n^2 + 128*n - 112)*x^2 - (1024*n^3 + 256*n^2 - 64*n - 80)*x^3 + (1024*n^3 - 192*n + 32)*x^4)*deriv(deriv(v[i - 2], x), x) + (32*n^2 - 32*n + 6 + (256*n^3 - 192*n^2 + 48*n - 4)*x)*v[i - 1] + ((128*n^2 - 24)*x - (256*n^3 + 64*n^2 - 80*n + 12)*x^2)*deriv(v[i - 1], x))/(16*n^2 - 8*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
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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.