Real-rooted atlas

A110441 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(6\,n\right)\,P_{n}(x) = \left(-12\,x - 8\,x^{2} + 18\,n + 36\,n x + 8\,n x^{2}\right)\,P_{n-1}(x) + \left(-9\,x - 24\,x^{2} - 8\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
29241447
32710813248315
4814328106241642111
5243162041044896272456014147
6729583218630302402613611328191294807
721872041278732162000191160128736456006528635355
865616998431638678926411842201090368599472179328222884257871
9196832361961224720359251265522527683984577756826784006929767609628534387
105904978732046058221553644833394032476824324570020028964736115935842641536259808191224967
11177147259815616927380645602401598784482694384003147258242543270401390845604894214499596168870401281505995
1253144185030566105666625981560072866466014167560961955143008192651955213431614406452352002024602563721651230285448588092031
13159432327634932216827928101847715231914016207035806448112126965121306125388811106110880679969728029101524488236062721380324481034009657553632227
1447829698928208875996063039057370561352600013633353394624603239860088120103552081653062176609704962563324292358412820958208330360192050871705635303296385699241527
Technical details
Node id
n-db00947034f68c16 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-5fc9190b1f03dd89

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:     3,      4;
  2:     9,     24,      14;
  3:    27,    108,     132,      48;
  4:    81,    432,     810,     624,     164;
  5:   243,   1620,    4104,    4896,    2724,     560;
  6:   729,   5832,   18630,   30240,   26136,   11328,    1912;
  7:  2187,  20412,   78732,  162000,  191160,  128736,   45600,    6528;
  8:  6561,  69984,  316386,  789264, 1184220, 1090368,  599472,  179328,  22288;
  9: 19683, 236196, 1224720, 3592512, 6552252, 7683984, 5777568, 2678400, 692976, 76096;
%F Formula
The row polynomials satisfy 6*n*P(n,x) = (18*n+(36*n-12)*x+(8*n-8)*x^2)*P(n-1,x) - (9*x+24*x^2+8*x^3)*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[((18*n + (36*n - 12)*x + (8*n - 8)*x^2)*P[n - 1, x] - (9*x + 24*x^2 + 8*x^3)*D[P[n - 1, x], x])/(6*n)]];
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] = ((18*n + (36*n - 12)*x + (8*n - 8)*x^2)*v[i - 1] - (9*x + 24*x^2 + 8*x^3)*deriv(v[i - 1], x))/(6*n)); 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

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