A154986 → θ+1 → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(3 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(-12\,x^{2} + 40\,n x^{2} - 63\,n^{2} x^{2} + 59\,n^{3} x^{2} - 32\,n^{4} x^{2} + 9\,n^{5} x^{2} - n^{6} x^{2}\right)\,P_{n-4}(x) + \left(-8\,x - 4\,x^{2} + 20\,n x + 10\,n x^{2} - 20\,n^{2} x - 12\,n^{2} x^{2} + 10\,n^{3} x + 8\,n^{3} x^{2} - 2\,n^{4} x - 2\,n^{4} x^{2}\right)\,P_{n-3}(x) + \left(-3 - 2\,x - x^{2} + 3\,n - 4\,n x + n x^{2} - n^{2} + 10\,n^{2} x - n^{2} x^{2} - 8\,n^{3} x + 2\,n^{4} x\right)\,P_{n-2}(x) + \left(6 + 4\,x - 6\,n - 4\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 3 + 8\,x + x^{2}\)
\(P_{3}(x) = 4 + 33\,x + 22\,x^{2} + x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 1 | 3 | ||||||
| 2 | 3 | 8 | 1 | 12 | |||||
| 3 | 4 | 33 | 22 | 1 | 60 | ||||
| 4 | 5 | 96 | 210 | 48 | 1 | 360 | |||
| 5 | 6 | 225 | 1256 | 942 | 90 | 1 | 2520 | ||
| 6 | 7 | 456 | 5395 | 10912 | 3237 | 152 | 1 | 20160 | |
| 7 | 8 | 833 | 18270 | 84975 | 67980 | 9135 | 238 | 1 | 181440 |
Compute more
?
Technical details
- Node id
n-e9df25e24f0e0fe4· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-c9cad0c5368eab69
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, 1; 2: 3, 8, 1; 3: 4, 33, 22, 1; 4: 5, 96, 210, 48, 1; 5: 6, 225, 1256, 942, 90, 1; 6: 7, 456, 5395, 10912, 3237, 152, 1; 7: 8, 833, 18270, 84975, 67980, 9135, 238, 1; 8: 9, 1408, 51940, 482784, 933790, 321856, 22260, 352, 1; 9: 10, 2241, 129312, 2149868, 8945172, 7454310, 1228496, 48492, 498, 1;
%F FormulaThe row polynomials satisfy (n^2-3*n+3)*P(n,x) = -(n^6-9*n^5+32*n^4-59*n^3+63*n^2-40*n+12)*x^2*P(n-4,x) - ((2*n^4-10*n^3+20*n^2-20*n+8)*x+(2*n^4-8*n^3+12*n^2-10*n+4)*x^2)*P(n-3,x) - (n^2-3*n+3-(2*n^4-8*n^3+10*n^2-4*n-2)*x+(n^2-n+1)*x^2)*P(n-2,x) + (2*n^2-6*n+6+(2*n^2-4*n+4)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 3 + 8*x + x^2, P(3,x) = 4 + 33*x + 22*x^2 + x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 3 + 8*x + x^2;
P[3, x_] := 4 + 33*x + 22*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2*P[n - 4, x] - ((2*n^4 - 10*n^3 + 20*n^2 - 20*n + 8)*x + (2*n^4 - 8*n^3 + 12*n^2 - 10*n + 4)*x^2)*P[n - 3, x] - (n^2 - 3*n + 3 - (2*n^4 - 8*n^3 + 10*n^2 - 4*n - 2)*x + (n^2 - n + 1)*x^2)*P[n - 2, x] + (2*n^2 - 6*n + 6 + (2*n^2 - 4*n + 4)*x)*P[n - 1, x])/(n^2 - 3*n + 3)]];
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 + x; v[3] = 3 + 8*x + x^2; v[4] = 4 + 33*x + 22*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2*v[i - 4] - ((2*n^4 - 10*n^3 + 20*n^2 - 20*n + 8)*x + (2*n^4 - 8*n^3 + 12*n^2 - 10*n + 4)*x^2)*v[i - 3] - (n^2 - 3*n + 3 - (2*n^4 - 8*n^3 + 10*n^2 - 4*n - 2)*x + (n^2 - n + 1)*x^2)*v[i - 2] + (2*n^2 - 6*n + 6 + (2*n^2 - 4*n + 4)*x)*v[i - 1])/(n^2 - 3*n + 3)); 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
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
theta_plus_one of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read…?
-
Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
-
from
Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.?
-
Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
from
Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.?
-
Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
-
-
from
theta_plus_one of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read…?
Related ?
Same fingerprint ?
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.