A274498 → D → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-8 + 26\,x - 90\,x^{2} + 8\,x^{3} - 8\,x^{4} + 4\,n - 33\,n x + 145\,n x^{2} - 54\,n x^{3} + 28\,n x^{4} + 10\,n^{2} x - 78\,n^{2} x^{2} + 75\,n^{2} x^{3} - 38\,n^{2} x^{4} + 14\,n^{3} x^{2} - 37\,n^{3} x^{3} + 25\,n^{3} x^{4} + 6\,n^{4} x^{3} - 8\,n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(-96 + 816\,x - 1872\,x^{2} + 816\,x^{3} - 96\,x^{4} + 60\,n - 993\,n x + 3345\,n x^{2} - 2280\,n x^{3} + 372\,n x^{4} - 8\,n^{2} + 348\,n^{2} x - 2136\,n^{2} x^{2} + 2505\,n^{2} x^{3} - 586\,n^{2} x^{4} - 34\,n^{3} x + 532\,n^{3} x^{2} - 1265\,n^{3} x^{3} + 527\,n^{3} x^{4} - 6\,n^{3} x^{5} - 40\,n^{4} x^{2} + 266\,n^{4} x^{3} - 246\,n^{4} x^{4} + 14\,n^{4} x^{5} - 16\,n^{5} x^{3} + 47\,n^{5} x^{4} - 10\,n^{5} x^{5} - 2\,n^{6} x^{4} + 2\,n^{6} x^{5}\right)\,P_{n-1}(x) + \left(-6 + 66\,x - 114\,x^{2} - 372\,x^{3} + 312\,x^{4} - 48\,x^{5} + 2\,n - 66\,n x + 268\,n x^{2} + 406\,n x^{3} - 732\,n x^{4} + 176\,n x^{5} + 12\,n^{2} x - 166\,n^{2} x^{2} - 2\,n^{2} x^{3} + 628\,n^{2} x^{4} - 256\,n^{2} x^{5} + 22\,n^{3} x^{2} - 94\,n^{3} x^{3} - 176\,n^{3} x^{4} + 198\,n^{3} x^{5} - 4\,n^{3} x^{6} + 12\,n^{4} x^{3} - 10\,n^{4} x^{4} - 64\,n^{4} x^{5} + 8\,n^{4} x^{6} + 2\,n^{5} x^{4} + 2\,n^{5} x^{5} - 4\,n^{5} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 6\)
Rows ?
| n | x0 | x1 | x2 | Pn(1) |
|---|---|---|---|---|
| 0 | 0 | |||
| 1 | 0 | |||
| 2 | 6 | 6 | ||
| 3 | 18 | 18 | ||
| 4 | 36 | 288 | 324 | |
| 5 | 108 | 864 | 972 | |
| 6 | 162 | 2592 | 5832 | 8586 |
| 7 | 486 | 7776 | 17496 | 25758 |
Compute more
?
Technical details
- Node id
n-4513125922bbf129· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-5367d60e6d4311ec
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: 6; 3: 18; 4: 36, 288; 5: 108, 864; 6: 162, 2592, 5832; 7: 486, 7776, 17496; 8: 648, 15552, 69984, 82944; 9: 1944, 46656, 209952, 248832;
%F FormulaThe row polynomials satisfy (4*n-8+(10*n^2-33*n+26)*x+(14*n^3-78*n^2+145*n-90)*x^2+(6*n^4-37*n^3+75*n^2-54*n+8)*x^3+(n^5-8*n^4+25*n^3-38*n^2+28*n-8)*x^4)*P(n,x) = -(8*n^2-60*n+96+(34*n^3-348*n^2+993*n-816)*x+(40*n^4-532*n^3+2136*n^2-3345*n+1872)*x^2+(16*n^5-266*n^4+1265*n^3-2505*n^2+2280*n-816)*x^3+(2*n^6-47*n^5+246*n^4-527*n^3+586*n^2-372*n+96)*x^4-(2*n^6-10*n^5+14*n^4-6*n^3)*x^5)*P(n-1,x) + (2*n-6+(12*n^2-66*n+66)*x+(22*n^3-166*n^2+268*n-114)*x^2+(12*n^4-94*n^3-2*n^2+406*n-372)*x^3+(2*n^5-10*n^4-176*n^3+628*n^2-732*n+312)*x^4+(2*n^5-64*n^4+198*n^3-256*n^2+176*n-48)*x^5-(4*n^5-8*n^4+4*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) = 6.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^2 - 60*n + 96 + (34*n^3 - 348*n^2 + 993*n - 816)*x + (40*n^4 - 532*n^3 + 2136*n^2 - 3345*n + 1872)*x^2 + (16*n^5 - 266*n^4 + 1265*n^3 - 2505*n^2 + 2280*n - 816)*x^3 + (2*n^6 - 47*n^5 + 246*n^4 - 527*n^3 + 586*n^2 - 372*n + 96)*x^4 - (2*n^6 - 10*n^5 + 14*n^4 - 6*n^3)*x^5)*P[n - 1, x] + (2*n - 6 + (12*n^2 - 66*n + 66)*x + (22*n^3 - 166*n^2 + 268*n - 114)*x^2 + (12*n^4 - 94*n^3 - 2*n^2 + 406*n - 372)*x^3 + (2*n^5 - 10*n^4 - 176*n^3 + 628*n^2 - 732*n + 312)*x^4 + (2*n^5 - 64*n^4 + 198*n^3 - 256*n^2 + 176*n - 48)*x^5 - (4*n^5 - 8*n^4 + 4*n^3)*x^6)*D[P[n - 1, x], x])/(4*n - 8 + (10*n^2 - 33*n + 26)*x + (14*n^3 - 78*n^2 + 145*n - 90)*x^2 + (6*n^4 - 37*n^3 + 75*n^2 - 54*n + 8)*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] = 6; for(n = 3, N, my(i = n + 1); v[i] = (-(8*n^2 - 60*n + 96 + (34*n^3 - 348*n^2 + 993*n - 816)*x + (40*n^4 - 532*n^3 + 2136*n^2 - 3345*n + 1872)*x^2 + (16*n^5 - 266*n^4 + 1265*n^3 - 2505*n^2 + 2280*n - 816)*x^3 + (2*n^6 - 47*n^5 + 246*n^4 - 527*n^3 + 586*n^2 - 372*n + 96)*x^4 - (2*n^6 - 10*n^5 + 14*n^4 - 6*n^3)*x^5)*v[i - 1] + (2*n - 6 + (12*n^2 - 66*n + 66)*x + (22*n^3 - 166*n^2 + 268*n - 114)*x^2 + (12*n^4 - 94*n^3 - 2*n^2 + 406*n - 372)*x^3 + (2*n^5 - 10*n^4 - 176*n^3 + 628*n^2 - 732*n + 312)*x^4 + (2*n^5 - 64*n^4 + 198*n^3 - 256*n^2 + 176*n - 48)*x^5 - (4*n^5 - 8*n^4 + 4*n^3)*x^6)*deriv(v[i - 1], x))/(4*n - 8 + (10*n^2 - 33*n + 26)*x + (14*n^3 - 78*n^2 + 145*n - 90)*x^2 + (6*n^4 - 37*n^3 + 75*n^2 - 54*n + 8)*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.