A334218 → θ+1 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(\left(-12 + 16\,n - 7\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(24 + 36\,x + 12\,x^{2} - 74\,n - 114\,n x - 40\,n x^{2} + 85\,n^{2} + 136\,n^{2} x + 51\,n^{2} x^{2} - 45\,n^{3} - 76\,n^{3} x - 31\,n^{3} x^{2} + 11\,n^{4} + 20\,n^{4} x + 9\,n^{4} x^{2} - n^{5} - 2\,n^{5} x - n^{5} x^{2}\right)\,P_{n-2}(x) + \left(6\,x + 10\,x^{2} + 4\,x^{3} - 17\,n x - 29\,n x^{2} - 12\,n x^{3} + 17\,n^{2} x + 30\,n^{2} x^{2} + 13\,n^{2} x^{3} - 7\,n^{3} x - 13\,n^{3} x^{2} - 6\,n^{3} x^{3} + n^{4} x + 2\,n^{4} x^{2} + n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(27 + 24\,x - 63\,n - 56\,n x + 51\,n^{2} + 46\,n^{2} x - 17\,n^{3} - 16\,n^{3} x + 2\,n^{4} + 2\,n^{4} x\right)\,P_{n-1}(x) + \left(6\,x + 6\,x^{2} - 11\,n x - 11\,n x^{2} + 6\,n^{2} x + 6\,n^{2} x^{2} - n^{3} x - n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 4 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||
| 2 | 1 | 1 | ||||||
| 3 | 4 | 4 | 8 | |||||
| 4 | 15 | 24 | 9 | 48 | ||||
| 5 | 72 | 152 | 96 | 16 | 336 | |||
| 6 | 420 | 1100 | 975 | 320 | 25 | 2840 | ||
| 7 | 2880 | 9000 | 10260 | 5064 | 960 | 36 | 28200 | |
| 8 | 22680 | 82320 | 114660 | 75656 | 23275 | 2688 | 49 | 321328 |
Compute more
?
Technical details
- Node id
n-9b67fd9b7ecdcab9· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-efe4fe8c256eb4ca
Observed on the stored rows ?
- Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
- 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 1; 3: 4, 4; 4: 15, 24, 9; 5: 72, 152, 96, 16; 6: 420, 1100, 975, 320, 25; 7: 2880, 9000, 10260, 5064, 960, 36; 8: 22680, 82320, 114660, 75656, 23275, 2688, 49; 9: 201600, 833280, 1370880, 1132992, 485520, 98832, 7168, 64; 10: 1995840, 9253440, 17554320, 17436384, 9604980, 2844180, 397215, 18432, 81;
%F FormulaThe row polynomials satisfy (n^3-7*n^2+16*n-12)*P(n,x) = -(n^5-11*n^4+45*n^3-85*n^2+74*n-24+(2*n^5-20*n^4+76*n^3-136*n^2+114*n-36)*x+(n^5-9*n^4+31*n^3-51*n^2+40*n-12)*x^2)*P(n-2,x) + ((n^4-7*n^3+17*n^2-17*n+6)*x+(2*n^4-13*n^3+30*n^2-29*n+10)*x^2+(n^4-6*n^3+13*n^2-12*n+4)*x^3)*d/dx P(n-2,x) + (2*n^4-17*n^3+51*n^2-63*n+27+(2*n^4-16*n^3+46*n^2-56*n+24)*x)*P(n-1,x) - ((n^3-6*n^2+11*n-6)*x+(n^3-6*n^2+11*n-6)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 4 + 4*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 11*n^4 + 45*n^3 - 85*n^2 + 74*n - 24 + (2*n^5 - 20*n^4 + 76*n^3 - 136*n^2 + 114*n - 36)*x + (n^5 - 9*n^4 + 31*n^3 - 51*n^2 + 40*n - 12)*x^2)*P[n - 2, x] + ((n^4 - 7*n^3 + 17*n^2 - 17*n + 6)*x + (2*n^4 - 13*n^3 + 30*n^2 - 29*n + 10)*x^2 + (n^4 - 6*n^3 + 13*n^2 - 12*n + 4)*x^3)*D[P[n - 2, x], x] + (2*n^4 - 17*n^3 + 51*n^2 - 63*n + 27 + (2*n^4 - 16*n^3 + 46*n^2 - 56*n + 24)*x)*P[n - 1, x] - ((n^3 - 6*n^2 + 11*n - 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*D[P[n - 1, x], x])/(n^3 - 7*n^2 + 16*n - 12)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 4 + 4*x; for(n = 4, N, my(i = n); v[i] = (-(n^5 - 11*n^4 + 45*n^3 - 85*n^2 + 74*n - 24 + (2*n^5 - 20*n^4 + 76*n^3 - 136*n^2 + 114*n - 36)*x + (n^5 - 9*n^4 + 31*n^3 - 51*n^2 + 40*n - 12)*x^2)*v[i - 2] + ((n^4 - 7*n^3 + 17*n^2 - 17*n + 6)*x + (2*n^4 - 13*n^3 + 30*n^2 - 29*n + 10)*x^2 + (n^4 - 6*n^3 + 13*n^2 - 12*n + 4)*x^3)*deriv(v[i - 2], x) + (2*n^4 - 17*n^3 + 51*n^2 - 63*n + 27 + (2*n^4 - 16*n^3 + 46*n^2 - 56*n + 24)*x)*v[i - 1] - ((n^3 - 6*n^2 + 11*n - 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*deriv(v[i - 1], x))/(n^3 - 7*n^2 + 16*n - 12)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, -1]
- Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
theta_plus_one of Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly…?
-
Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
-
from
Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.?
-
Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
-
-
from
Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.?
-
-
from
theta_plus_one of Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly…?
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.