A257615 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(9 + 12\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-225 - 348\,x + 369\,x^{2} - 240\,n - 1016\,n x + 1392\,n x^{2} - 60\,n^{2} - 720\,n^{2} x + 1688\,n^{2} x^{2} - 160\,n^{3} x + 832\,n^{3} x^{2} + 144\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-384\,x + 1284\,x^{2} - 900\,x^{3} - 320\,n x + 1832\,n x^{2} - 1272\,n x^{3} - 80\,n^{2} x + 944\,n^{2} x^{2} - 352\,n^{2} x^{3} + 160\,n^{3} x^{2} + 160\,n^{3} x^{3} + 64\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-84\,x^{2} + 408\,x^{3} - 324\,x^{4} - 64\,n x^{2} + 640\,n x^{3} - 576\,n x^{4} - 16\,n^{2} x^{2} + 352\,n^{2} x^{3} - 336\,n^{2} x^{4} + 64\,n^{3} x^{3} - 64\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(90 + 108\,x + 108\,n + 216\,n x + 32\,n^{2} + 144\,n^{2} x + 32\,n^{3} x\right)\,P_{n-1}(x) + \left(60\,x - 180\,x^{2} + 56\,n x - 312\,n x^{2} + 16\,n^{2} x - 176\,n^{2} x^{2} - 32\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 20\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 5 | 20 | 25 | ||||||
| 2 | 25 | 280 | 225 | 530 | |||||
| 3 | 125 | 2860 | 6435 | 2000 | 11420 | ||||
| 4 | 625 | 25520 | 115830 | 102080 | 15625 | 259680 | |||
| 5 | 3125 | 211140 | 1674090 | 2976160 | 1319625 | 112500 | 6296640 | ||
| 6 | 15625 | 1665480 | 21242655 | 65475520 | 59007375 | 14989320 | 765625 | 163161600 | |
| 7 | 78125 | 12720860 | 247393845 | 1211172080 | 1892456375 | 989575380 | 155830535 | 5000000 | 4514227200 |
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Technical details
- Node id
n-41fe3e2d4341ffdb· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-ea23bfbe9c2dc6f1
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: 5, 20; 2: 25, 280, 225; 3: 125, 2860, 6435, 2000; 4: 625, 25520, 115830, 102080, 15625; 5: 3125, 211140, 1674090, 2976160, 1319625, 112500; 6: 15625, 1665480, 21242655, 65475520, 59007375, 14989320, 765625; 7: 78125, 12720860, 247393845, 1211172080, 1892456375, 989575380, 155830535, 5000000; 8: 390625, 94983520, 2713117500, 19920062080, 49203865750, 44820139680, 14771417500, 1519736320, 31640625; 9: 1953125, 697697140, 28478602980, 301117122880, 1106526709750, 1593398462040, 922171188820, 202514510080, 14128367085, 195312500;
%F FormulaThe row polynomials satisfy (4*n^2+12*n+9)*P(n,x) = -(60*n^2+240*n+225+(160*n^3+720*n^2+1016*n+348)*x-(144*n^4+832*n^3+1688*n^2+1392*n+369)*x^2)*P(n-2,x) - ((80*n^2+320*n+384)*x-(160*n^3+944*n^2+1832*n+1284)*x^2-(64*n^4+160*n^3-352*n^2-1272*n-900)*x^3)*d/dx P(n-2,x) - ((16*n^2+64*n+84)*x^2-(64*n^3+352*n^2+640*n+408)*x^3+(64*n^3+336*n^2+576*n+324)*x^4)*d^2/dx^2 P(n-2,x) + (32*n^2+108*n+90+(32*n^3+144*n^2+216*n+108)*x)*P(n-1,x) + ((16*n^2+56*n+60)*x-(32*n^3+176*n^2+312*n+180)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 20*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 5 + 20*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(60*n^2 + 240*n + 225 + (160*n^3 + 720*n^2 + 1016*n + 348)*x - (144*n^4 + 832*n^3 + 1688*n^2 + 1392*n + 369)*x^2)*P[n - 2, x] - ((80*n^2 + 320*n + 384)*x - (160*n^3 + 944*n^2 + 1832*n + 1284)*x^2 - (64*n^4 + 160*n^3 - 352*n^2 - 1272*n - 900)*x^3)*D[P[n - 2, x], x] - ((16*n^2 + 64*n + 84)*x^2 - (64*n^3 + 352*n^2 + 640*n + 408)*x^3 + (64*n^3 + 336*n^2 + 576*n + 324)*x^4)*D[P[n - 2, x], {x, 2}] + (32*n^2 + 108*n + 90 + (32*n^3 + 144*n^2 + 216*n + 108)*x)*P[n - 1, x] + ((16*n^2 + 56*n + 60)*x - (32*n^3 + 176*n^2 + 312*n + 180)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 12*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] = 5 + 20*x; for(n = 2, N, my(i = n + 1); v[i] = (-(60*n^2 + 240*n + 225 + (160*n^3 + 720*n^2 + 1016*n + 348)*x - (144*n^4 + 832*n^3 + 1688*n^2 + 1392*n + 369)*x^2)*v[i - 2] - ((80*n^2 + 320*n + 384)*x - (160*n^3 + 944*n^2 + 1832*n + 1284)*x^2 - (64*n^4 + 160*n^3 - 352*n^2 - 1272*n - 900)*x^3)*deriv(v[i - 2], x) - ((16*n^2 + 64*n + 84)*x^2 - (64*n^3 + 352*n^2 + 640*n + 408)*x^3 + (64*n^3 + 336*n^2 + 576*n + 324)*x^4)*deriv(deriv(v[i - 2], x), x) + (32*n^2 + 108*n + 90 + (32*n^3 + 144*n^2 + 216*n + 108)*x)*v[i - 1] + ((16*n^2 + 56*n + 60)*x - (32*n^3 + 176*n^2 + 312*n + 180)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 12*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]
-
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
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.