Real-rooted atlas

A172431 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(6\,n + 12\,n x + 6\,n x^{2}\right)\,P_{n}(x) = \left(-11 - 83\,x - 201\,x^{2} - 197\,x^{3} - 68\,x^{4} + 29\,n + 173\,n x + 327\,n x^{2} + 251\,n x^{3} + 68\,n x^{4}\right)\,P_{n-1}(x) + \left(-5 - 52\,x - 194\,x^{2} - 320\,x^{3} - 241\,x^{4} - 68\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13912
284665119
3211774894411128
45561024994468292910561
514419801074628710376471930598532
6377618641799148624292674301818126881918359
7987188191521816757051776375275967923402138330498557008
825845613852859528147529260020192452182463354117716928546734579725121
967651649551772502110073544349591711329387519426794021114303613174658735877321742776252
10177114789405783058410328601893183815929834681275693090185913182417541876609661416382354183696920186999
1146368137714418459252147319452777336102284543695873673768701348064472617065571481142179752617006565889154472818564472837688
1212139339281165787409851313126030477731851276827951638662408302851959139761354841852851514983170101129499209835035166439610135859761600669377281
13317811111287971787570461743232890115102611455431356020718833349038448560276563293042702243013074197558701308535104366882519422562359109679143665070868895596211585172
148320403134743054518899757995381844214107061222117246577086424949762525530220577045726055928053969759604657412199699694305110464501025606799855929830254479486453043639002582552137804590039
Technical details
Node id
n-4f0975de88203a4c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e8d55f52162be113

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,      9;
  2:    8,     46,      65;
  3:   21,    177,     489,      441;
  4:   55,    610,    2499,     4468,     2929;
  5:  144,   1980,   10746,    28710,    37647,     19305;
  6:  377,   6186,   41799,   148624,   292674,    301818,    126881;
  7:  987,  18819,  152181,   675705,  1776375,   2759679,   2340213,    833049;
  8: 2584,  56138,  528595,  2814752,  9260020,  19245218,  24633541,  17716928,   5467345;
  9: 6765, 164955, 1772502, 11007354, 43495917, 113293875, 194267940, 211143036, 131746587, 35877321;
%F Formula
The row polynomials satisfy (6*n+12*n*x+6*n*x^2)*P(n,x) = (29*n-11+(173*n-83)*x+(327*n-201)*x^2+(251*n-197)*x^3+(68*n-68)*x^4)*P(n-1,x) - (5+52*x+194*x^2+320*x^3+241*x^4+68*x^5)*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[((29*n - 11 + (173*n - 83)*x + (327*n - 201)*x^2 + (251*n - 197)*x^3 + (68*n - 68)*x^4)*P[n - 1, x] - (5 + 52*x + 194*x^2 + 320*x^3 + 241*x^4 + 68*x^5)*D[P[n - 1, x], x])/(6*n + 12*n*x + 6*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] = ((29*n - 11 + (173*n - 83)*x + (327*n - 201)*x^2 + (251*n - 197)*x^3 + (68*n - 68)*x^4)*v[i - 1] - (5 + 52*x + 194*x^2 + 320*x^3 + 241*x^4 + 68*x^5)*deriv(v[i - 1], x))/(6*n + 12*n*x + 6*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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023