Real-rooted atlas

A207606 → basis A013610 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-594 + 1188\,x - 486\,x^{3} - 426\,n + 2304\,n x - 54\,n x^{2} - 324\,n x^{3} + 1458\,n^{2} x + 54\,n^{3} + 324\,n^{3} x\right)\,P_{n}(x) = \left(-2364 + 2628\,x + 4374\,x^{2} - 1782\,x^{3} - 972\,x^{4} - 1116\,n + 6030\,n x + 6939\,n x^{2} - 1053\,n x^{3} - 486\,n x^{4} + 108\,n^{2} + 3402\,n^{2} x + 3402\,n^{2} x^{2} + 81\,n^{3} + 567\,n^{3} x + 486\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-220 - 177\,x + 1233\,x^{2} - 1269\,x^{3} - 2025\,x^{4} - 486\,x^{5} + 45\,n + 1242\,n x + 3969\,n x^{2} + 1944\,n x^{3} + 45\,n^{2} + 432\,n^{2} x + 1053\,n^{2} x^{2} + 486\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 46 + 69\,x + 18\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0466918133
116643831554973
25542139270012691626824
3176390361671313716477948646493
45434348488580610297862370172531458310147
5163691262823902856234305508002638986050743742035945
648464437421163116032726973844260268053310614242077651312213196846
7141566146422264051651552273222895946211978621219106741115607020273936684671513

All 15 rows

Technical details
Node id
n-6a266bbd778ea6ef · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d8d0ec869029bda3

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 Example
Triangle begins:
  0:      46,       69,       18;
  1:     166,      438,      315,        54;
  2:     554,     2139,     2700,      1269,       162;
  3:    1763,     9036,    16713,     13716,      4779,       486;
  4:    5434,    34848,    85806,    102978,     62370,     17253,      1458;
  5:   16369,   126282,   390285,    623430,    550800,    263898,     60507,      4374;
  6:   48464,   437421,  1631160,   3272697,   3844260,   2680533,   1061424,    207765,     13122;
  7:  141566,  1464222,  6405165,  15522732,  22895946,  21197862,  12191067,   4111560,    702027,    39366;
  8:  409072,  4771803, 23977404,  68249358, 121619232, 141296967, 107754948,  52689204,  15470838,  2342277,  118098;
  9: 1171621, 15219648, 86429538, 282939264, 592451415, 831373956, 794975958, 514828548, 218835594, 56883870, 7735419, 354294;
%F Formula
The row polynomials satisfy (54*n^3-426*n-594+(324*n^3+1458*n^2+2304*n+1188)*x-54*n*x^2-(324*n+486)*x^3)*P(n,x) = (81*n^3+108*n^2-1116*n-2364+(567*n^3+3402*n^2+6030*n+2628)*x+(486*n^3+3402*n^2+6939*n+4374)*x^2-(1053*n+1782)*x^3-(486*n+972)*x^4)*P(n-1,x) + (45*n^2+45*n-220+(432*n^2+1242*n-177)*x+(1053*n^2+3969*n+1233)*x^2+(486*n^2+1944*n-1269)*x^3-2025*x^4-486*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 46 + 69*x + 18*x^2.
%t Mathematica
P[0, x_] := 46 + 69*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((81*n^3 + 108*n^2 - 1116*n - 2364 + (567*n^3 + 3402*n^2 + 6030*n + 2628)*x + (486*n^3 + 3402*n^2 + 6939*n + 4374)*x^2 - (1053*n + 1782)*x^3 - (486*n + 972)*x^4)*P[n - 1, x] + (45*n^2 + 45*n - 220 + (432*n^2 + 1242*n - 177)*x + (1053*n^2 + 3969*n + 1233)*x^2 + (486*n^2 + 1944*n - 1269)*x^3 - 2025*x^4 - 486*x^5)*D[P[n - 1, x], x])/(54*n^3 - 426*n - 594 + (324*n^3 + 1458*n^2 + 2304*n + 1188)*x - 54*n*x^2 - (324*n + 486)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 46 + 69*x + 18*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((81*n^3 + 108*n^2 - 1116*n - 2364 + (567*n^3 + 3402*n^2 + 6030*n + 2628)*x + (486*n^3 + 3402*n^2 + 6939*n + 4374)*x^2 - (1053*n + 1782)*x^3 - (486*n + 972)*x^4)*v[i - 1] + (45*n^2 + 45*n - 220 + (432*n^2 + 1242*n - 177)*x + (1053*n^2 + 3969*n + 1233)*x^2 + (486*n^2 + 1944*n - 1269)*x^3 - 2025*x^4 - 486*x^5)*deriv(v[i - 1], x))/(54*n^3 - 426*n - 594 + (324*n^3 + 1458*n^2 + 2304*n + 1188)*x - 54*n*x^2 - (324*n + 486)*x^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

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023