Real-rooted atlas

A249250 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 3\,x - 3\,x^{2} - x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(41 + 299\,x + 887\,x^{2} + 1377\,x^{3} + 1184\,x^{4} + 536\,x^{5} + 100\,x^{6} - 68\,n - 479\,n x - 1387\,n x^{2} - 2117\,n x^{3} - 1799\,n x^{4} - 808\,n x^{5} - 150\,n x^{6} + 27\,n^{2} + 180\,n^{2} x + 500\,n^{2} x^{2} + 740\,n^{2} x^{3} + 615\,n^{2} x^{4} + 272\,n^{2} x^{5} + 50\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(66 + 561\,x + 2014\,x^{2} + 3966\,x^{3} + 4634\,x^{4} + 3217\,x^{5} + 1230\,x^{6} + 200\,x^{7} - 36\,n - 300\,n x - 1059\,n x^{2} - 2056\,n x^{3} - 2374\,n x^{4} - 1632\,n x^{5} - 619\,n x^{6} - 100\,n x^{7}\right)\,P_{n-1}'(x) + \left(12 + 120\,x + 517\,x^{2} + 1255\,x^{3} + 1880\,x^{4} + 1782\,x^{5} + 1045\,x^{6} + 347\,x^{7} + 50\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 7\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14711
210322567
328132204103467
476472108610964093139
52081604490074085540163921299
65685232199204011245042267366553144163
71552166247579219061628554825474812530826215976403
84240517762749608292801553380185031213685505746161048576611971
9115841588329627843385920761256011346012112093327078152259208441943144776691
10316484814083279600131756163456516061867728765091926454608035550234115427201677721303227107
1186464144492810928320493739041480489443093420004595665444854160803572291161744401165086852467108872053455827
12236224430195235765664179487232605522016144665971225096444803185150016293512018819151387448398700942222645522684354513906004419
1364537612721216115309248636368896238556032064143587521272664896018864832896208909113121706955730010002450628398053276896422016410737418394171492019
141763200373998083670700802209441792911125740827229798912608177094721031244184321333960397041309934472169612462656051112722064186167934104157528336429496729637729513123
Technical details
Node id
n-6ea4d72dc1454e17 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9e00e04d10cd46cf

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,      7;
  2:    10,     32,     25;
  3:    28,    132,    204,     103;
  4:    76,    472,   1086,    1096,     409;
  5:   208,   1604,   4900,    7408,    5540,     1639;
  6:   568,   5232,  19920,   40112,   45042,    26736,     6553;
  7:  1552,  16624,  75792,  190616,  285548,   254748,   125308,   26215;
  8:  4240,  51776, 274960,  829280, 1553380,  1850312,  1368550,  574616,  104857;
  9: 11584, 158832, 962784, 3385920, 7612560, 11346012, 11209332, 7078152, 2592084, 419431;
%F Formula
The row polynomials satisfy (n-1+(3*n-3)*x+(3*n-3)*x^2+(n-1)*x^3)*P(n,x) = (27*n^2-68*n+41+(180*n^2-479*n+299)*x+(500*n^2-1387*n+887)*x^2+(740*n^2-2117*n+1377)*x^3+(615*n^2-1799*n+1184)*x^4+(272*n^2-808*n+536)*x^5+(50*n^2-150*n+100)*x^6)*P(n-1,x) - (36*n-66+(300*n-561)*x+(1059*n-2014)*x^2+(2056*n-3966)*x^3+(2374*n-4634)*x^4+(1632*n-3217)*x^5+(619*n-1230)*x^6+(100*n-200)*x^7)*d/dx P(n-1,x) + (12+120*x+517*x^2+1255*x^3+1880*x^4+1782*x^5+1045*x^6+347*x^7+50*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 7*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 7*x;
P[n_, x_] := P[n, x] = Expand[Together[((27*n^2 - 68*n + 41 + (180*n^2 - 479*n + 299)*x + (500*n^2 - 1387*n + 887)*x^2 + (740*n^2 - 2117*n + 1377)*x^3 + (615*n^2 - 1799*n + 1184)*x^4 + (272*n^2 - 808*n + 536)*x^5 + (50*n^2 - 150*n + 100)*x^6)*P[n - 1, x] - (36*n - 66 + (300*n - 561)*x + (1059*n - 2014)*x^2 + (2056*n - 3966)*x^3 + (2374*n - 4634)*x^4 + (1632*n - 3217)*x^5 + (619*n - 1230)*x^6 + (100*n - 200)*x^7)*D[P[n - 1, x], x] + (12 + 120*x + 517*x^2 + 1255*x^3 + 1880*x^4 + 1782*x^5 + 1045*x^6 + 347*x^7 + 50*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 + 7*x; for(n = 2, N, my(i = n + 1); v[i] = ((27*n^2 - 68*n + 41 + (180*n^2 - 479*n + 299)*x + (500*n^2 - 1387*n + 887)*x^2 + (740*n^2 - 2117*n + 1377)*x^3 + (615*n^2 - 1799*n + 1184)*x^4 + (272*n^2 - 808*n + 536)*x^5 + (50*n^2 - 150*n + 100)*x^6)*v[i - 1] - (36*n - 66 + (300*n - 561)*x + (1059*n - 2014)*x^2 + (2056*n - 3966)*x^3 + (2374*n - 4634)*x^4 + (1632*n - 3217)*x^5 + (619*n - 1230)*x^6 + (100*n - 200)*x^7)*deriv(v[i - 1], x) + (12 + 120*x + 517*x^2 + 1255*x^3 + 1880*x^4 + 1782*x^5 + 1045*x^6 + 347*x^7 + 50*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