Real-rooted atlas

A113413 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(9 + 27\,x + 27\,x^{2} + 9\,x^{3} + 9\,n + 27\,n x + 27\,n x^{2} + 9\,n x^{3}\right)\,P_{n}(x) = \left(217 + 1467\,x + 3999\,x^{2} + 5693\,x^{3} + 4500\,x^{4} + 1884\,x^{5} + 328\,x^{6} - 213\,n - 1557\,n x - 4617\,n x^{2} - 7131\,n x^{3} - 6066\,n x^{4} - 2700\,n x^{5} - 492\,n x^{6} + 50\,n^{2} + 360\,n^{2} x + 1104\,n^{2} x^{2} + 1816\,n^{2} x^{3} + 1674\,n^{2} x^{4} + 816\,n^{2} x^{5} + 164\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(116 + 1199\,x + 4938\,x^{2} + 10738\,x^{3} + 13492\,x^{4} + 9879\,x^{5} + 3926\,x^{6} + 656\,x^{7} - 40\,n - 424\,n x - 1821\,n x^{2} - 4172\,n x^{3} - 5558\,n x^{4} - 4332\,n x^{5} - 1837\,n x^{6} - 328\,n x^{7}\right)\,P_{n-1}'(x) + \left(8 + 112\,x + 641\,x^{2} + 1969\,x^{3} + 3590\,x^{4} + 4018\,x^{5} + 2717\,x^{6} + 1021\,x^{7} + 164\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
27323473
317120270194601
4413921362203611064937
599119456221290614442630640569
6239348020700643361101489849635954333353
75779844707882779646434368778726536862049942739161
813932724822985210916243191320587950466657584252172116878622507657
9336374190718260400378814157168329225165034582648818046272381466663906184945209
10811919940821791341394875257902124162833472314132790410464200347659200172371160379946741519693033
11196015304166457970466850222225971567349010721714020018282382959032203510802421165140108011726221662899412487303321
1247321139884018776670151301948814981842309092702484624540421685022130824216361470244966096401655654749867133023081235123666102608053577
131142433662706537359944775837822868327570122913901983865845720690346246794156879408582199893265530181663490478111511217970414403728907042134306843129408249
1427580795320721517456721474628192977117231246695462272165950881932445510979232907807335312139716287619215986983501841318822366688741464660828254314960624401511661946927986393513
Technical details
Node id
n-4f15ce4a317311f6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0fd5c3ec633d3157

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,     6;
  2:    7,    32,     34;
  3:   17,   120,    270,     194;
  4:   41,   392,   1362,    2036,     1106;
  5:   99,  1194,   5622,   12906,    14442,     6306;
  6:  239,  3480,  20700,   64336,   110148,    98496,    35954;
  7:  577,  9844,  70788,  277964,   643436,   877872,   653686,   204994;
  8: 1393, 27248, 229852, 1091624,  3191320,  5879504,  6665758,  4252172,  1168786;
  9: 3363, 74190, 718260, 4003788, 14157168, 32922516, 50345826, 48818046, 27238146, 6663906;
%F Formula
The row polynomials satisfy (9*n+9+(27*n+27)*x+(27*n+27)*x^2+(9*n+9)*x^3)*P(n,x) = (50*n^2-213*n+217+(360*n^2-1557*n+1467)*x+(1104*n^2-4617*n+3999)*x^2+(1816*n^2-7131*n+5693)*x^3+(1674*n^2-6066*n+4500)*x^4+(816*n^2-2700*n+1884)*x^5+(164*n^2-492*n+328)*x^6)*P(n-1,x) - (40*n-116+(424*n-1199)*x+(1821*n-4938)*x^2+(4172*n-10738)*x^3+(5558*n-13492)*x^4+(4332*n-9879)*x^5+(1837*n-3926)*x^6+(328*n-656)*x^7)*d/dx P(n-1,x) + (8+112*x+641*x^2+1969*x^3+3590*x^4+4018*x^5+2717*x^6+1021*x^7+164*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[((50*n^2 - 213*n + 217 + (360*n^2 - 1557*n + 1467)*x + (1104*n^2 - 4617*n + 3999)*x^2 + (1816*n^2 - 7131*n + 5693)*x^3 + (1674*n^2 - 6066*n + 4500)*x^4 + (816*n^2 - 2700*n + 1884)*x^5 + (164*n^2 - 492*n + 328)*x^6)*P[n - 1, x] - (40*n - 116 + (424*n - 1199)*x + (1821*n - 4938)*x^2 + (4172*n - 10738)*x^3 + (5558*n - 13492)*x^4 + (4332*n - 9879)*x^5 + (1837*n - 3926)*x^6 + (328*n - 656)*x^7)*D[P[n - 1, x], x] + (8 + 112*x + 641*x^2 + 1969*x^3 + 3590*x^4 + 4018*x^5 + 2717*x^6 + 1021*x^7 + 164*x^8)*D[P[n - 1, x], {x, 2}])/(9*n + 9 + (27*n + 27)*x + (27*n + 27)*x^2 + (9*n + 9)*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] = ((50*n^2 - 213*n + 217 + (360*n^2 - 1557*n + 1467)*x + (1104*n^2 - 4617*n + 3999)*x^2 + (1816*n^2 - 7131*n + 5693)*x^3 + (1674*n^2 - 6066*n + 4500)*x^4 + (816*n^2 - 2700*n + 1884)*x^5 + (164*n^2 - 492*n + 328)*x^6)*v[i - 1] - (40*n - 116 + (424*n - 1199)*x + (1821*n - 4938)*x^2 + (4172*n - 10738)*x^3 + (5558*n - 13492)*x^4 + (4332*n - 9879)*x^5 + (1837*n - 3926)*x^6 + (328*n - 656)*x^7)*deriv(v[i - 1], x) + (8 + 112*x + 641*x^2 + 1969*x^3 + 3590*x^4 + 4018*x^5 + 2717*x^6 + 1021*x^7 + 164*x^8)*deriv(deriv(v[i - 1], x), x))/(9*n + 9 + (27*n + 27)*x + (27*n + 27)*x^2 + (9*n + 9)*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