Real-rooted atlas

A207824 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(78\,n - 54\,n x - 54\,n x^{2} + 54\,n^{3}\right)\,P_{n}(x) = \left(108 - 288\,x - 324\,x^{2} - 81\,x^{3} + 432\,n - 126\,n x - 324\,n x^{2} - 81\,n x^{3} + 486\,n^{2} + 243\,n^{2} x + 162\,n^{3} + 81\,n^{3} x\right)\,P_{n-1}(x) + \left(-448 - 1368\,x - 1386\,x^{2} - 567\,x^{3} - 81\,x^{4} + 288\,n + 324\,n x + 81\,n x^{2} + 288\,n^{2} + 324\,n^{2} x + 81\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19312
271549134
3522642243271434
43673631838619728114905
5250835573747628193323645243151668
616761545781250549428674087075131227291518587
71101780357565248546543557790150903036595845927218715012978
8714931326900532434133183912397221433815725646614645611574646561146906002
945910413196607103368011836394277652265595841116108559327000245651208531441196831425413760
10292304807140421292229925198633719242332297851620516008315125820073561403004242119859117714705904913732964531
11184775196698426657342353752504933311996145309493450801977431551288372733402749400172579270690777478505845851177147131513314536
12116092165696792328369818016050307528616954292030278648151022443611608811922790042845589134744123934993652318854230279951309191318765314411252914147253
13725577713914626021119251348041557160237539853612028211023885502381508871290146767202791266701913112822282630256753782911765904937734899273178862178597159432311882617491864
1445142374585531153496821209894804853500191587513431602524447111077023921696179908167897899994588851621302216351424393778710421749328762261681807223457129993234750926992008846984782969112248747328670
Technical details
Node id
n-3bc2d94af59781a9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a5c5a90bfc57f410

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 Example
Triangle begins:
  0:        1;
  1:        9,         3;
  2:       71,        54,         9;
  3:      522,       642,       243,        27;
  4:     3673,      6318,      3861,       972,        81;
  5:    25083,     55737,     47628,     19332,      3645,       243;
  6:   167615,    457812,    505494,    286740,     87075,     13122,      729;
  7:  1101780,   3575652,   4854654,   3557790,   1509030,    365958,    45927,    2187;
  8:  7149313,  26900532,  43413318,  39123972,  21433815,   7256466,  1464561,  157464,   6561;
  9: 45910413, 196607103, 368011836, 394277652, 265595841, 116108559, 32700024, 5651208, 531441, 19683;
%F Formula
The row polynomials satisfy (54*n^3+78*n-54*n*x-54*n*x^2)*P(n,x) = (162*n^3+486*n^2+432*n+108+(81*n^3+243*n^2-126*n-288)*x-(324*n+324)*x^2-(81*n+81)*x^3)*P(n-1,x) + (288*n^2+288*n-448+(324*n^2+324*n-1368)*x+(81*n^2+81*n-1386)*x^2-567*x^3-81*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((162*n^3 + 486*n^2 + 432*n + 108 + (81*n^3 + 243*n^2 - 126*n - 288)*x - (324*n + 324)*x^2 - (81*n + 81)*x^3)*P[n - 1, x] + (288*n^2 + 288*n - 448 + (324*n^2 + 324*n - 1368)*x + (81*n^2 + 81*n - 1386)*x^2 - 567*x^3 - 81*x^4)*D[P[n - 1, x], x])/(54*n^3 + 78*n - 54*n*x - 54*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((162*n^3 + 486*n^2 + 432*n + 108 + (81*n^3 + 243*n^2 - 126*n - 288)*x - (324*n + 324)*x^2 - (81*n + 81)*x^3)*v[i - 1] + (288*n^2 + 288*n - 448 + (324*n^2 + 324*n - 1368)*x + (81*n^2 + 81*n - 1386)*x^2 - 567*x^3 - 81*x^4)*deriv(v[i - 1], x))/(54*n^3 + 78*n - 54*n*x - 54*n*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

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