Real-rooted atlas

A086645 → basis A013611 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-32 - 24\,x + 16\,n + 12\,n x\right)\,P_{n-1}(x) + \left(-10 - 18\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
26410
3566416136
4576960480642080
560161331210240307225632896
66297617408018176087040179201024524800
76594562187264288768019087366451209830440968390656
8690585626722304426639363601203217031168435814451609616384134225920
972318976319815680598999040615514112374865920135790592275251202621440655362147516416
10757334016376779571280982835209809166336731499724834374942729964093441651507201297612826214434359869440
1179309045764384096256010634657792014836301824013097500672075753324544286733107206857687040951582720629145601048576549756338176
1283053510656505019367424136462729216021547523768322198098411520151081346662470712400281622215131136044845498368530579456029989273641943048796095119360
13869747654656576941614694417184331923456302956926730243507043801497627937487192064155935163023366083485630464162191219097628125324902428789702656140928614416777216140737496743936
14910811568537665452952780800213043799654400414798546206720537255110246400486825147760640315792503603200147394553446400489904773529601127596687360017033907404801526726656006543114240671088642251799847239680
1595381425750016738157775552512260666262814720055549925890457607959911208058880808595776077824059794883661004803251746583347200129692375973888037349840912384075237089607680100153268633607938978611203006477107226843545636028797153181696

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-16 - 32\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(12 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 4\,x\)
Technical details
Node id
n-0f17d4387a7e32e2 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-037adb803e01af7d)
Fingerprint
f6-f83cf33d2c000fbc

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 -1)
  • 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 Example
Triangle begins:
   1:         1;
   2:         6,          4;
   3:        56,         64,         16;
   4:       576,        960,        480,         64;
   5:      6016,      13312,      10240,       3072,        256;
   6:     62976,     174080,     181760,      87040,      17920,       1024;
   7:    659456,    2187264,    2887680,    1908736,     645120,      98304,      4096;
   8:   6905856,   26722304,   42663936,   36012032,   17031168,    4358144,    516096,     16384;
   9:  72318976,  319815680,  598999040,  615514112,  374865920,  135790592,  27525120,   2621440,    65536;
  10: 757334016, 3767795712, 8098283520, 9809166336, 7314997248, 3437494272, 996409344, 165150720, 12976128, 262144;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (16*n-32+(12*n-24)*x)*P(n-1,x) - (10+18*x+8*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n - 32 + (12*n - 24)*x)*P[n - 1, x] - (10 + 18*x + 8*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 4*x; for(n = 3, N, my(i = n); v[i] = ((16*n - 32 + (12*n - 24)*x)*v[i - 1] - (10 + 18*x + 8*x^2)*deriv(v[i - 1], x))/(n - 2)); 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

Proof chains ? JSON

Same fingerprint ?

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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023