A110441 → reverse → θ+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(2\,n + 4\,n^{2} + 6\,n^{2} x - 2\,n^{2} x^{2} + 12\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-6\,x - 12\,x^{2} - 6\,x^{3} + 2\,n + 6\,n x - 21\,n x^{2} - 18\,n x^{3} - n x^{4} + 4\,n^{2} + 42\,n^{2} x + 85\,n^{2} x^{2} + 9\,n^{2} x^{3} + 12\,n^{3} x + 56\,n^{3} x^{2} + 15\,n^{3} x^{3} + n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-x - 9\,x^{2} - 19\,x^{3} - 3\,x^{4} - 2\,n x - 24\,n x^{2} - 75\,n x^{3} - 18\,n x^{4} - n x^{5} - 6\,n^{2} x^{2} - 37\,n^{2} x^{3} - 12\,n^{2} x^{4} - n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 1 | 6 | 7 | |||||||||||||
| 2 | 1 | 12 | 21 | 34 | ||||||||||||
| 3 | 1 | 18 | 69 | 60 | 148 | |||||||||||
| 4 | 1 | 24 | 144 | 288 | 155 | 612 | ||||||||||
| 5 | 1 | 30 | 246 | 792 | 1005 | 378 | 2452 | |||||||||
| 6 | 1 | 36 | 375 | 1680 | 3495 | 3132 | 889 | 9608 | ||||||||
| 7 | 1 | 42 | 531 | 3060 | 8975 | 13338 | 9037 | 2040 | 37024 | |||||||
| 8 | 1 | 48 | 714 | 5040 | 19200 | 40608 | 45934 | 24672 | 4599 | 140816 | ||||||
| 9 | 1 | 54 | 924 | 7728 | 36330 | 100548 | 162932 | 146592 | 64629 | 10230 | 529968 | |||||
| 10 | 1 | 60 | 1161 | 11232 | 62930 | 216216 | 461090 | 596928 | 441261 | 163980 | 22517 | 1977376 | ||||
| 11 | 1 | 66 | 1425 | 15660 | 101970 | 419580 | 1116122 | 1909680 | 2036925 | 1268370 | 405669 | 49140 | 7324608 | |||
| 12 | 1 | 72 | 1716 | 21120 | 156825 | 752976 | 2408448 | 5159808 | 7296435 | 6565800 | 3512652 | 983232 | 106483 | 26965568 | ||
| 13 | 1 | 78 | 2034 | 27720 | 231275 | 1270566 | 4758348 | 12298464 | 21840759 | 26113050 | 20200642 | 9435096 | 2343133 | 229362 | 98750528 | |
| 14 | 1 | 84 | 2379 | 35568 | 329505 | 2039796 | 8766219 | 26604864 | 57154923 | 86016060 | 88537977 | 59788368 | 24704563 | 5505276 | 491505 | 359977088 |
Compute more
?
Technical details
- Node id
n-275d0e075f661270· revision 0 · JSON- Definition source
derived:theta_plus_one- Fingerprint
f6-61571462e09ff891
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: 1, 6; 2: 1, 12, 21; 3: 1, 18, 69, 60; 4: 1, 24, 144, 288, 155; 5: 1, 30, 246, 792, 1005, 378; 6: 1, 36, 375, 1680, 3495, 3132, 889; 7: 1, 42, 531, 3060, 8975, 13338, 9037, 2040; 8: 1, 48, 714, 5040, 19200, 40608, 45934, 24672, 4599; 9: 1, 54, 924, 7728, 36330, 100548, 162932, 146592, 64629, 10230;
%F FormulaThe row polynomials satisfy (4*n^2+2*n+(12*n^3+6*n^2)*x+(2*n^3-2*n^2)*x^2)*P(n,x) = (4*n^2+2*n+(12*n^3+42*n^2+6*n-6)*x+(56*n^3+85*n^2-21*n-12)*x^2+(15*n^3+9*n^2-18*n-6)*x^3+(n^3-n)*x^4)*P(n-1,x) - ((2*n+1)*x+(6*n^2+24*n+9)*x^2+(37*n^2+75*n+19)*x^3+(12*n^2+18*n+3)*x^4+(n^2+n)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 2*n + (12*n^3 + 42*n^2 + 6*n - 6)*x + (56*n^3 + 85*n^2 - 21*n - 12)*x^2 + (15*n^3 + 9*n^2 - 18*n - 6)*x^3 + (n^3 - n)*x^4)*P[n - 1, x] - ((2*n + 1)*x + (6*n^2 + 24*n + 9)*x^2 + (37*n^2 + 75*n + 19)*x^3 + (12*n^2 + 18*n + 3)*x^4 + (n^2 + n)*x^5)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (12*n^3 + 6*n^2)*x + (2*n^3 - 2*n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 2*n + (12*n^3 + 42*n^2 + 6*n - 6)*x + (56*n^3 + 85*n^2 - 21*n - 12)*x^2 + (15*n^3 + 9*n^2 - 18*n - 6)*x^3 + (n^3 - n)*x^4)*v[i - 1] - ((2*n + 1)*x + (6*n^2 + 24*n + 9)*x^2 + (37*n^2 + 75*n + 19)*x^3 + (12*n^2 + 18*n + 3)*x^4 + (n^2 + n)*x^5)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (12*n^3 + 6*n^2)*x + (2*n^3 - 2*n^2)*x^2)); 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
-
Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
-
from
reversal of Triangular array formed by the Mersenne numbers.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangular array formed by the Mersenne numbers.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangular array formed by the Mersenne numbers.?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
-
from
reversal of Triangular array formed by the Mersenne numbers.?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
- [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.
- [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023