Real-rooted atlas

A116412 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(16 + 48\,x + 48\,x^{2} + 16\,x^{3} + 16\,n + 48\,n x + 48\,n x^{2} + 16\,n x^{3}\right)\,P_{n}(x) = \left(446 + 3048\,x + 8358\,x^{2} + 11936\,x^{3} + 9450\,x^{4} + 3960\,x^{5} + 690\,x^{6} - 431\,n - 3190\,n x - 9549\,n x^{2} - 14852\,n x^{3} - 12697\,n x^{4} - 5670\,n x^{5} - 1035\,n x^{6} + 113\,n^{2} + 782\,n^{2} x + 2343\,n^{2} x^{2} + 3812\,n^{2} x^{3} + 3503\,n^{2} x^{4} + 1710\,n^{2} x^{5} + 345\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(216 + 2324\,x + 9824\,x^{2} + 21756\,x^{3} + 27704\,x^{4} + 20492\,x^{5} + 8208\,x^{6} + 1380\,x^{7} - 78\,n - 838\,n x - 3646\,n x^{2} - 8454\,n x^{3} - 11386\,n x^{4} - 8962\,n x^{5} - 3834\,n x^{6} - 690\,n x^{7}\right)\,P_{n-1}'(x) + \left(13 + 196\,x + 1184\,x^{2} + 3780\,x^{3} + 7090\,x^{4} + 8108\,x^{5} + 5576\,x^{6} + 2124\,x^{7} + 345\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
2135861132
3432976574671464
4142132845246656357416224
5469553325522575306340127353179808
61549220621284993916846590435805902093411992768
75116853766009962312872525510870504805173276160215222085376
8168973232962669236124185763560707064430464718612844517838412261769244766976
95580712042391140961262286108215888562492686370740348796706458348388513287938431432712694272
1018431844280444735200629677608012071331483329444808630633039681003150246753386790330058497271821084630064146432
1160876116113439191966407135855725763454852742053771474247001014670760572408188529137289363132677051277643526355496691637333193795584
12201060158135596763535154602246495631770906279118087665240317085712188619764346584875161733319870760122236579510451890231652733292420678956893692707713024
136640564208247000298924548026003063840152926634380642062752344197982552273645396449276167740475517196969357961844086668694907285239277555808191957473310832195872800840925402677248
1421932293741424150115478671751098282637407125247218053335500125386116183820696393059043038791261178117418663924846197307621040355456531658444843515833246763203644871158135703563742464050574501453566519328768
Technical details
Node id
n-c090ab59d4a9c3de · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2f094be7a82a92fc

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,       8;
  2:    13,      58,       61;
  3:    43,     297,      657,      467;
  4:   142,    1328,     4524,     6656,      3574;
  5:   469,    5533,    25522,    57530,     63401,     27353;
  6:  1549,   22062,   128499,   391684,    659043,    580590,    209341;
  7:  5116,   85376,   600996,  2312872,   5255108,   7050480,   5173276,   1602152;
  8: 16897,  323296,  2669236, 12418576,  35607070,  64430464,  71861284,  45178384,  12261769;
  9: 55807, 1204239, 11409612, 62286108, 215888562, 492686370, 740348796, 706458348, 388513287, 93843143;
%F Formula
The row polynomials satisfy (16*n+16+(48*n+48)*x+(48*n+48)*x^2+(16*n+16)*x^3)*P(n,x) = (113*n^2-431*n+446+(782*n^2-3190*n+3048)*x+(2343*n^2-9549*n+8358)*x^2+(3812*n^2-14852*n+11936)*x^3+(3503*n^2-12697*n+9450)*x^4+(1710*n^2-5670*n+3960)*x^5+(345*n^2-1035*n+690)*x^6)*P(n-1,x) - (78*n-216+(838*n-2324)*x+(3646*n-9824)*x^2+(8454*n-21756)*x^3+(11386*n-27704)*x^4+(8962*n-20492)*x^5+(3834*n-8208)*x^6+(690*n-1380)*x^7)*d/dx P(n-1,x) + (13+196*x+1184*x^2+3780*x^3+7090*x^4+8108*x^5+5576*x^6+2124*x^7+345*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[((113*n^2 - 431*n + 446 + (782*n^2 - 3190*n + 3048)*x + (2343*n^2 - 9549*n + 8358)*x^2 + (3812*n^2 - 14852*n + 11936)*x^3 + (3503*n^2 - 12697*n + 9450)*x^4 + (1710*n^2 - 5670*n + 3960)*x^5 + (345*n^2 - 1035*n + 690)*x^6)*P[n - 1, x] - (78*n - 216 + (838*n - 2324)*x + (3646*n - 9824)*x^2 + (8454*n - 21756)*x^3 + (11386*n - 27704)*x^4 + (8962*n - 20492)*x^5 + (3834*n - 8208)*x^6 + (690*n - 1380)*x^7)*D[P[n - 1, x], x] + (13 + 196*x + 1184*x^2 + 3780*x^3 + 7090*x^4 + 8108*x^5 + 5576*x^6 + 2124*x^7 + 345*x^8)*D[P[n - 1, x], {x, 2}])/(16*n + 16 + (48*n + 48)*x + (48*n + 48)*x^2 + (16*n + 16)*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] = ((113*n^2 - 431*n + 446 + (782*n^2 - 3190*n + 3048)*x + (2343*n^2 - 9549*n + 8358)*x^2 + (3812*n^2 - 14852*n + 11936)*x^3 + (3503*n^2 - 12697*n + 9450)*x^4 + (1710*n^2 - 5670*n + 3960)*x^5 + (345*n^2 - 1035*n + 690)*x^6)*v[i - 1] - (78*n - 216 + (838*n - 2324)*x + (3646*n - 9824)*x^2 + (8454*n - 21756)*x^3 + (11386*n - 27704)*x^4 + (8962*n - 20492)*x^5 + (3834*n - 8208)*x^6 + (690*n - 1380)*x^7)*deriv(v[i - 1], x) + (13 + 196*x + 1184*x^2 + 3780*x^3 + 7090*x^4 + 8108*x^5 + 5576*x^6 + 2124*x^7 + 345*x^8)*deriv(deriv(v[i - 1], x), x))/(16*n + 16 + (48*n + 48)*x + (48*n + 48)*x^2 + (16*n + 16)*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