Real-rooted atlas

A081577 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(28 + 200\,x + 580\,x^{2} + 880\,x^{3} + 740\,x^{4} + 328\,x^{5} + 60\,x^{6} - 38\,n - 274\,n x - 808\,n x^{2} - 1250\,n x^{3} - 1072\,n x^{4} - 484\,n x^{5} - 90\,n x^{6} + 12\,n^{2} + 84\,n^{2} x + 246\,n^{2} x^{2} + 384\,n^{2} x^{3} + 336\,n^{2} x^{4} + 156\,n^{2} x^{5} + 30\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(27 + 255\,x + 990\,x^{2} + 2070\,x^{3} + 2535\,x^{4} + 1827\,x^{5} + 720\,x^{6} + 120\,x^{7} - 12\,n - 114\,n x - 448\,n x^{2} - 952\,n x^{3} - 1188\,n x^{4} - 874\,n x^{5} - 352\,n x^{6} - 60\,n x^{7}\right)\,P_{n-1}'(x) + \left(3 + 36\,x + 181\,x^{2} + 500\,x^{3} + 835\,x^{4} + 868\,x^{5} + 551\,x^{6} + 196\,x^{7} + 30\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
26242252
31696184112408
444352102412805803280
51201200468889288280299226208
632839361930449472698165142415448209728
78961254474048238720453632507904310208797441677696
82448391682703361050624251337637885443513216183244841166413421824
96688120384951168432768012490560237077762958604823406336106523202125120107374080
1018272365440325257616958976573456001313656322064182402196587521514921926114752010970464858993664
114992010982401087244863915264247836672665419776126207667216907381761567691008958170624347445888566325766871947264
121363843273216356761602334515201021212864314543616069936675841130864025613197098560108392903685947648512195769753629235308854975582208
133726089687808115275520831007232404846195214072748288359036113926806255923295862505600990692733447301886131236347845120109532322561509207808439804649472
14101798428503552367658624289506816015545047680602006287361733763175683771788021766228005728007766957798407201070273284813034608642192300225286091635456077909497603518437212160
Technical details
Node id
n-e32ebd201e4ce3f4 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c3ea4df52af7b3cb

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:    2,      4;
  2:    6,     24,     22;
  3:   16,     96,    184,     112;
  4:   44,    352,   1024,    1280,      580;
  5:  120,   1200,   4688,    8928,     8280,     2992;
  6:  328,   3936,  19304,   49472,    69816,    51424,    15448;
  7:  896,  12544,  74048,  238720,   453632,   507904,   310208,    79744;
  8: 2448,  39168, 270336, 1050624,  2513376,  3788544,  3513216,  1832448,   411664;
  9: 6688, 120384, 951168, 4327680, 12490560, 23707776, 29586048, 23406336, 10652320, 2125120;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (12*n^2-38*n+28+(84*n^2-274*n+200)*x+(246*n^2-808*n+580)*x^2+(384*n^2-1250*n+880)*x^3+(336*n^2-1072*n+740)*x^4+(156*n^2-484*n+328)*x^5+(30*n^2-90*n+60)*x^6)*P(n-1,x) - (12*n-27+(114*n-255)*x+(448*n-990)*x^2+(952*n-2070)*x^3+(1188*n-2535)*x^4+(874*n-1827)*x^5+(352*n-720)*x^6+(60*n-120)*x^7)*d/dx P(n-1,x) + (3+36*x+181*x^2+500*x^3+835*x^4+868*x^5+551*x^6+196*x^7+30*x^8)*d^2/dx^2 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[((12*n^2 - 38*n + 28 + (84*n^2 - 274*n + 200)*x + (246*n^2 - 808*n + 580)*x^2 + (384*n^2 - 1250*n + 880)*x^3 + (336*n^2 - 1072*n + 740)*x^4 + (156*n^2 - 484*n + 328)*x^5 + (30*n^2 - 90*n + 60)*x^6)*P[n - 1, x] - (12*n - 27 + (114*n - 255)*x + (448*n - 990)*x^2 + (952*n - 2070)*x^3 + (1188*n - 2535)*x^4 + (874*n - 1827)*x^5 + (352*n - 720)*x^6 + (60*n - 120)*x^7)*D[P[n - 1, x], x] + (3 + 36*x + 181*x^2 + 500*x^3 + 835*x^4 + 868*x^5 + 551*x^6 + 196*x^7 + 30*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + n*x^3)]];
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] = ((12*n^2 - 38*n + 28 + (84*n^2 - 274*n + 200)*x + (246*n^2 - 808*n + 580)*x^2 + (384*n^2 - 1250*n + 880)*x^3 + (336*n^2 - 1072*n + 740)*x^4 + (156*n^2 - 484*n + 328)*x^5 + (30*n^2 - 90*n + 60)*x^6)*v[i - 1] - (12*n - 27 + (114*n - 255)*x + (448*n - 990)*x^2 + (952*n - 2070)*x^3 + (1188*n - 2535)*x^4 + (874*n - 1827)*x^5 + (352*n - 720)*x^6 + (60*n - 120)*x^7)*deriv(v[i - 1], x) + (3 + 36*x + 181*x^2 + 500*x^3 + 835*x^4 + 868*x^5 + 551*x^6 + 196*x^7 + 30*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + n*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. [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