A274496 → D → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-128 + 208\,x - 360\,x^{2} + 16\,x^{3} - 8\,x^{4} + 64\,n - 264\,n x + 580\,n x^{2} - 108\,n x^{3} + 28\,n x^{4} + 80\,n^{2} x - 312\,n^{2} x^{2} + 150\,n^{2} x^{3} - 38\,n^{2} x^{4} + 56\,n^{3} x^{2} - 74\,n^{3} x^{3} + 25\,n^{3} x^{4} + 12\,n^{4} x^{3} - 8\,n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(-1024 + 4352\,x - 4992\,x^{2} + 1088\,x^{3} - 64\,x^{4} + 576\,n - 5224\,n x + 8980\,n x^{2} - 3112\,n x^{3} + 252\,n x^{4} - 64\,n^{2} + 1696\,n^{2} x - 5696\,n^{2} x^{2} + 3506\,n^{2} x^{3} - 406\,n^{2} x^{4} - 136\,n^{3} x + 1332\,n^{3} x^{2} - 1780\,n^{3} x^{3} + 389\,n^{3} x^{4} - 3\,n^{3} x^{5} - 80\,n^{4} x^{2} + 354\,n^{4} x^{3} - 189\,n^{4} x^{4} + 7\,n^{4} x^{5} - 16\,n^{5} x^{3} + 35\,n^{5} x^{4} - 5\,n^{5} x^{5} - n^{6} x^{4} + n^{6} x^{5}\right)\,P_{n-1}(x) + \left(-128 + 704\,x - 608\,x^{2} - 992\,x^{3} + 416\,x^{4} - 32\,x^{5} + 32\,n - 688\,n x + 1464\,n x^{2} + 1064\,n x^{3} - 1000\,n x^{4} + 120\,n x^{5} + 96\,n^{2} x - 888\,n^{2} x^{2} + 76\,n^{2} x^{3} + 880\,n^{2} x^{4} - 180\,n^{2} x^{5} + 88\,n^{3} x^{2} - 292\,n^{3} x^{3} - 230\,n^{3} x^{4} + 148\,n^{3} x^{5} - 2\,n^{3} x^{6} + 24\,n^{4} x^{3} - 28\,n^{4} x^{4} - 48\,n^{4} x^{5} + 4\,n^{4} x^{6} + 2\,n^{5} x^{4} - 2\,n^{5} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)
Rows ?
| n | x0 | x1 | x2 | Pn(1) |
|---|---|---|---|---|
| 0 | 0 | |||
| 1 | 0 | |||
| 2 | 2 | 2 | ||
| 3 | 4 | 4 | ||
| 4 | 8 | 32 | 40 | |
| 5 | 16 | 64 | 80 | |
| 6 | 24 | 192 | 216 | 432 |
| 7 | 48 | 384 | 432 | 864 |
Compute more
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Technical details
- Node id
n-c846bc324cb25f0c· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-4f7c344a4c25ac22
Observed on the stored rows ?
- Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
- 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: 0; 1: 0; 2: 2; 3: 4; 4: 8, 32; 5: 16, 64; 6: 24, 192, 216; 7: 48, 384, 432; 8: 64, 768, 1728, 1024; 9: 128, 1536, 3456, 2048;
%F FormulaThe row polynomials satisfy (64*n-128+(80*n^2-264*n+208)*x+(56*n^3-312*n^2+580*n-360)*x^2+(12*n^4-74*n^3+150*n^2-108*n+16)*x^3+(n^5-8*n^4+25*n^3-38*n^2+28*n-8)*x^4)*P(n,x) = -(64*n^2-576*n+1024+(136*n^3-1696*n^2+5224*n-4352)*x+(80*n^4-1332*n^3+5696*n^2-8980*n+4992)*x^2+(16*n^5-354*n^4+1780*n^3-3506*n^2+3112*n-1088)*x^3+(n^6-35*n^5+189*n^4-389*n^3+406*n^2-252*n+64)*x^4-(n^6-5*n^5+7*n^4-3*n^3)*x^5)*P(n-1,x) + (32*n-128+(96*n^2-688*n+704)*x+(88*n^3-888*n^2+1464*n-608)*x^2+(24*n^4-292*n^3+76*n^2+1064*n-992)*x^3+(2*n^5-28*n^4-230*n^3+880*n^2-1000*n+416)*x^4-(48*n^4-148*n^3+180*n^2-120*n+32)*x^5-(2*n^5-4*n^4+2*n^3)*x^6)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(64*n^2 - 576*n + 1024 + (136*n^3 - 1696*n^2 + 5224*n - 4352)*x + (80*n^4 - 1332*n^3 + 5696*n^2 - 8980*n + 4992)*x^2 + (16*n^5 - 354*n^4 + 1780*n^3 - 3506*n^2 + 3112*n - 1088)*x^3 + (n^6 - 35*n^5 + 189*n^4 - 389*n^3 + 406*n^2 - 252*n + 64)*x^4 - (n^6 - 5*n^5 + 7*n^4 - 3*n^3)*x^5)*P[n - 1, x] + (32*n - 128 + (96*n^2 - 688*n + 704)*x + (88*n^3 - 888*n^2 + 1464*n - 608)*x^2 + (24*n^4 - 292*n^3 + 76*n^2 + 1064*n - 992)*x^3 + (2*n^5 - 28*n^4 - 230*n^3 + 880*n^2 - 1000*n + 416)*x^4 - (48*n^4 - 148*n^3 + 180*n^2 - 120*n + 32)*x^5 - (2*n^5 - 4*n^4 + 2*n^3)*x^6)*D[P[n - 1, x], x])/(64*n - 128 + (80*n^2 - 264*n + 208)*x + (56*n^3 - 312*n^2 + 580*n - 360)*x^2 + (12*n^4 - 74*n^3 + 150*n^2 - 108*n + 16)*x^3 + (n^5 - 8*n^4 + 25*n^3 - 38*n^2 + 28*n - 8)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = (-(64*n^2 - 576*n + 1024 + (136*n^3 - 1696*n^2 + 5224*n - 4352)*x + (80*n^4 - 1332*n^3 + 5696*n^2 - 8980*n + 4992)*x^2 + (16*n^5 - 354*n^4 + 1780*n^3 - 3506*n^2 + 3112*n - 1088)*x^3 + (n^6 - 35*n^5 + 189*n^4 - 389*n^3 + 406*n^2 - 252*n + 64)*x^4 - (n^6 - 5*n^5 + 7*n^4 - 3*n^3)*x^5)*v[i - 1] + (32*n - 128 + (96*n^2 - 688*n + 704)*x + (88*n^3 - 888*n^2 + 1464*n - 608)*x^2 + (24*n^4 - 292*n^3 + 76*n^2 + 1064*n - 992)*x^3 + (2*n^5 - 28*n^4 - 230*n^3 + 880*n^2 - 1000*n + 416)*x^4 - (48*n^4 - 148*n^3 + 180*n^2 - 120*n + 32)*x^5 - (2*n^5 - 4*n^4 + 2*n^3)*x^6)*deriv(v[i - 1], x))/(64*n - 128 + (80*n^2 - 264*n + 208)*x + (56*n^3 - 312*n^2 + 580*n - 360)*x^2 + (12*n^4 - 74*n^3 + 150*n^2 - 108*n + 16)*x^3 + (n^5 - 8*n^4 + 25*n^3 - 38*n^2 + 28*n - 8)*x^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
Proof chains ? JSON
Related ?
Same fingerprint ?
References
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.