Real-rooted atlas

A249250 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 3\,x - 3\,x^{2} - x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(14 + 122\,x + 410\,x^{2} + 698\,x^{3} + 644\,x^{4} + 308\,x^{5} + 60\,x^{6} - 26\,n - 206\,n x - 656\,n x^{2} - 1082\,n x^{3} - 980\,n x^{4} - 464\,n x^{5} - 90\,n x^{6} + 12\,n^{2} + 84\,n^{2} x + 246\,n^{2} x^{2} + 384\,n^{2} x^{3} + 336\,n^{2} x^{4} + 156\,n^{2} x^{5} + 30\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(21 + 207\,x + 838\,x^{2} + 1822\,x^{3} + 2313\,x^{4} + 1723\,x^{5} + 700\,x^{6} + 120\,x^{7} - 12\,n - 114\,n x - 448\,n x^{2} - 952\,n x^{3} - 1188\,n x^{4} - 874\,n x^{5} - 352\,n x^{6} - 60\,n x^{7}\right)\,P_{n-1}'(x) + \left(3 + 36\,x + 181\,x^{2} + 500\,x^{3} + 835\,x^{4} + 868\,x^{5} + 551\,x^{6} + 196\,x^{7} + 30\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 10\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141014
2104446100
328180372244824
4766401968260812526544
5208216888481763217120647252448
65687056358809545613994410708833400419392
715522238413627245348889009610296486496481724323355520
84240696324936961972352485296075261447181056385318489012826843392
9115842134081726848805094423821056463651205934374448150528224703364595104214748672
1031648646336587721631321344108292800253724544407836992443987712313234848129311680237211841717986304
11864641938752195702401173486724642817281272224640246310694433685651203188627904198941881673627193612245536013743896576
122362245769216640104964265167361900377792596329574413510339584222625766402647690963222161777664123915640324155638016632148544109951160320
136453761705254420626713615119672327491596800264908977926876356352013264184524819010854425619993044544014994990054475949131264232841236483263326336879609307136
14176320050114816656336512524873574428628014976112636897024329620030080728725588992122290093939215500093601281460560390016992112318976459223557760129654163712168461966087036874407936
Technical details
Node id
n-e82344c5ef80ab5d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c6176a18b14d9443

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:     4,     10;
  2:    10,     44,      46;
  3:    28,    180,     372,     244;
  4:    76,    640,    1968,    2608,     1252;
  5:   208,   2168,    8848,   17632,    17120,     6472;
  6:   568,   7056,   35880,   95456,   139944,   107088,    33400;
  7:  1552,  22384,  136272,  453488,   890096,  1029648,   649648,   172432;
  8:  4240,  69632,  493696, 1972352,  4852960,  7526144,  7181056,  3853184,   890128;
  9: 11584, 213408, 1726848, 8050944, 23821056, 46365120, 59343744, 48150528, 22470336, 4595104;
%F Formula
The row polynomials satisfy (n-1+(3*n-3)*x+(3*n-3)*x^2+(n-1)*x^3)*P(n,x) = (12*n^2-26*n+14+(84*n^2-206*n+122)*x+(246*n^2-656*n+410)*x^2+(384*n^2-1082*n+698)*x^3+(336*n^2-980*n+644)*x^4+(156*n^2-464*n+308)*x^5+(30*n^2-90*n+60)*x^6)*P(n-1,x) - (12*n-21+(114*n-207)*x+(448*n-838)*x^2+(952*n-1822)*x^3+(1188*n-2313)*x^4+(874*n-1723)*x^5+(352*n-700)*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 >= 2, with P(0,x) = 1, P(1,x) = 4 + 10*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 10*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n^2 - 26*n + 14 + (84*n^2 - 206*n + 122)*x + (246*n^2 - 656*n + 410)*x^2 + (384*n^2 - 1082*n + 698)*x^3 + (336*n^2 - 980*n + 644)*x^4 + (156*n^2 - 464*n + 308)*x^5 + (30*n^2 - 90*n + 60)*x^6)*P[n - 1, x] - (12*n - 21 + (114*n - 207)*x + (448*n - 838)*x^2 + (952*n - 1822)*x^3 + (1188*n - 2313)*x^4 + (874*n - 1723)*x^5 + (352*n - 700)*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 - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*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; v[2] = 4 + 10*x; for(n = 2, N, my(i = n + 1); v[i] = ((12*n^2 - 26*n + 14 + (84*n^2 - 206*n + 122)*x + (246*n^2 - 656*n + 410)*x^2 + (384*n^2 - 1082*n + 698)*x^3 + (336*n^2 - 980*n + 644)*x^4 + (156*n^2 - 464*n + 308)*x^5 + (30*n^2 - 90*n + 60)*x^6)*v[i - 1] - (12*n - 21 + (114*n - 207)*x + (448*n - 838)*x^2 + (952*n - 1822)*x^3 + (1188*n - 2313)*x^4 + (874*n - 1723)*x^5 + (352*n - 700)*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 - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*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