Real-rooted atlas

A056939 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(24\,n + 72\,n x + 72\,n x^{2} + 24\,n x^{3} + 50\,n^{2} + 150\,n^{2} x + 150\,n^{2} x^{2} + 50\,n^{2} x^{3} + 35\,n^{3} + 105\,n^{3} x + 105\,n^{3} x^{2} + 35\,n^{3} x^{3} + 10\,n^{4} + 30\,n^{4} x + 30\,n^{4} x^{2} + 10\,n^{4} x^{3} + n^{5} + 3\,n^{5} x + 3\,n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n}(x) = \left(36\,x + 168\,x^{2} + 336\,x^{3} + 360\,x^{4} + 204\,x^{5} + 48\,x^{6} + 24\,n + 195\,n x + 542\,n x^{2} + 772\,n x^{3} + 654\,n x^{4} + 329\,n x^{5} + 76\,n x^{6} + 50\,n^{2} + 250\,n^{2} x + 329\,n^{2} x^{2} - 99\,n^{2} x^{3} - 521\,n^{2} x^{4} - 379\,n^{2} x^{5} - 86\,n^{2} x^{6} + 35\,n^{3} + 163\,n^{3} x + 187\,n^{3} x^{2} - 137\,n^{3} x^{3} - 428\,n^{3} x^{4} - 302\,n^{3} x^{5} - 70\,n^{3} x^{6} + 10\,n^{4} + 65\,n^{4} x + 169\,n^{4} x^{2} + 237\,n^{4} x^{3} + 197\,n^{4} x^{4} + 94\,n^{4} x^{5} + 20\,n^{4} x^{6} + n^{5} + 11\,n^{5} x + 45\,n^{5} x^{2} + 91\,n^{5} x^{3} + 98\,n^{5} x^{4} + 54\,n^{5} x^{5} + 12\,n^{5} x^{6}\right)\,P_{n-1}(x) + \left(36\,x + 228\,x^{2} + 648\,x^{3} + 1032\,x^{4} + 948\,x^{5} + 468\,x^{6} + 96\,x^{7} + 123\,n x + 715\,n x^{2} + 1878\,n x^{3} + 2822\,n x^{4} + 2503\,n x^{5} + 1215\,n x^{6} + 248\,n x^{7} + 108\,n^{2} x + 491\,n^{2} x^{2} + 960\,n^{2} x^{3} + 1090\,n^{2} x^{4} + 800\,n^{2} x^{5} + 363\,n^{2} x^{6} + 76\,n^{2} x^{7} + 33\,n^{3} x + 64\,n^{3} x^{2} - 138\,n^{3} x^{3} - 532\,n^{3} x^{4} - 623\,n^{3} x^{5} - 324\,n^{3} x^{6} - 64\,n^{3} x^{7} + 3\,n^{4} x - 12\,n^{4} x^{2} - 102\,n^{4} x^{3} - 228\,n^{4} x^{4} - 237\,n^{4} x^{5} - 120\,n^{4} x^{6} - 24\,n^{4} x^{7}\right)\,P_{n-1}'(x) + \left(12\,x^{2} + 84\,x^{3} + 240\,x^{4} + 360\,x^{5} + 300\,x^{6} + 132\,x^{7} + 24\,x^{8} + 37\,n x^{2} + 259\,n x^{3} + 740\,n x^{4} + 1110\,n x^{5} + 925\,n x^{6} + 407\,n x^{7} + 74\,n x^{8} + 28\,n^{2} x^{2} + 196\,n^{2} x^{3} + 560\,n^{2} x^{4} + 840\,n^{2} x^{5} + 700\,n^{2} x^{6} + 308\,n^{2} x^{7} + 56\,n^{2} x^{8} + 6\,n^{3} x^{2} + 42\,n^{3} x^{3} + 120\,n^{3} x^{4} + 180\,n^{3} x^{5} + 150\,n^{3} x^{6} + 66\,n^{3} x^{7} + 12\,n^{3} x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1134
21101223
31237355152
41442765042761101
51758152705343514778508
611182045109402416523442829669007
7117545573665912511120540116095148393581248

All 15 rows

Technical details
Node id
n-2500f91b4fa8e65f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-7430f58d15347800

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: 1,   3;
  2: 1,  10,    12;
  3: 1,  23,    73,     55;
  4: 1,  44,   276,    504,     276;
  5: 1,  75,   815,   2705,    3435,    1477;
  6: 1, 118,  2045,  10940,   24165,   23442,     8296;
  7: 1, 175,  4557,  36659,  125111,  205401,   160951,    48393;
  8: 1, 248,  9268, 106904,  527590, 1302728,  1695484,  1113464,  291010;
  9: 1, 339, 17526, 279594, 1910496, 6646920, 12771150, 13741170, 7763160, 1794320;
%F Formula
The row polynomials satisfy (n^5+10*n^4+35*n^3+50*n^2+24*n+(3*n^5+30*n^4+105*n^3+150*n^2+72*n)*x+(3*n^5+30*n^4+105*n^3+150*n^2+72*n)*x^2+(n^5+10*n^4+35*n^3+50*n^2+24*n)*x^3)*P(n,x) = (n^5+10*n^4+35*n^3+50*n^2+24*n+(11*n^5+65*n^4+163*n^3+250*n^2+195*n+36)*x+(45*n^5+169*n^4+187*n^3+329*n^2+542*n+168)*x^2+(91*n^5+237*n^4-137*n^3-99*n^2+772*n+336)*x^3+(98*n^5+197*n^4-428*n^3-521*n^2+654*n+360)*x^4+(54*n^5+94*n^4-302*n^3-379*n^2+329*n+204)*x^5+(12*n^5+20*n^4-70*n^3-86*n^2+76*n+48)*x^6)*P(n-1,x) + ((3*n^4+33*n^3+108*n^2+123*n+36)*x-(12*n^4-64*n^3-491*n^2-715*n-228)*x^2-(102*n^4+138*n^3-960*n^2-1878*n-648)*x^3-(228*n^4+532*n^3-1090*n^2-2822*n-1032)*x^4-(237*n^4+623*n^3-800*n^2-2503*n-948)*x^5-(120*n^4+324*n^3-363*n^2-1215*n-468)*x^6-(24*n^4+64*n^3-76*n^2-248*n-96)*x^7)*d/dx P(n-1,x) + ((6*n^3+28*n^2+37*n+12)*x^2+(42*n^3+196*n^2+259*n+84)*x^3+(120*n^3+560*n^2+740*n+240)*x^4+(180*n^3+840*n^2+1110*n+360)*x^5+(150*n^3+700*n^2+925*n+300)*x^6+(66*n^3+308*n^2+407*n+132)*x^7+(12*n^3+56*n^2+74*n+24)*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[((n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (11*n^5 + 65*n^4 + 163*n^3 + 250*n^2 + 195*n + 36)*x + (45*n^5 + 169*n^4 + 187*n^3 + 329*n^2 + 542*n + 168)*x^2 + (91*n^5 + 237*n^4 - 137*n^3 - 99*n^2 + 772*n + 336)*x^3 + (98*n^5 + 197*n^4 - 428*n^3 - 521*n^2 + 654*n + 360)*x^4 + (54*n^5 + 94*n^4 - 302*n^3 - 379*n^2 + 329*n + 204)*x^5 + (12*n^5 + 20*n^4 - 70*n^3 - 86*n^2 + 76*n + 48)*x^6)*P[n - 1, x] + ((3*n^4 + 33*n^3 + 108*n^2 + 123*n + 36)*x - (12*n^4 - 64*n^3 - 491*n^2 - 715*n - 228)*x^2 - (102*n^4 + 138*n^3 - 960*n^2 - 1878*n - 648)*x^3 - (228*n^4 + 532*n^3 - 1090*n^2 - 2822*n - 1032)*x^4 - (237*n^4 + 623*n^3 - 800*n^2 - 2503*n - 948)*x^5 - (120*n^4 + 324*n^3 - 363*n^2 - 1215*n - 468)*x^6 - (24*n^4 + 64*n^3 - 76*n^2 - 248*n - 96)*x^7)*D[P[n - 1, x], x] + ((6*n^3 + 28*n^2 + 37*n + 12)*x^2 + (42*n^3 + 196*n^2 + 259*n + 84)*x^3 + (120*n^3 + 560*n^2 + 740*n + 240)*x^4 + (180*n^3 + 840*n^2 + 1110*n + 360)*x^5 + (150*n^3 + 700*n^2 + 925*n + 300)*x^6 + (66*n^3 + 308*n^2 + 407*n + 132)*x^7 + (12*n^3 + 56*n^2 + 74*n + 24)*x^8)*D[P[n - 1, x], {x, 2}])/(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (3*n^5 + 30*n^4 + 105*n^3 + 150*n^2 + 72*n)*x + (3*n^5 + 30*n^4 + 105*n^3 + 150*n^2 + 72*n)*x^2 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*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] = ((n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (11*n^5 + 65*n^4 + 163*n^3 + 250*n^2 + 195*n + 36)*x + (45*n^5 + 169*n^4 + 187*n^3 + 329*n^2 + 542*n + 168)*x^2 + (91*n^5 + 237*n^4 - 137*n^3 - 99*n^2 + 772*n + 336)*x^3 + (98*n^5 + 197*n^4 - 428*n^3 - 521*n^2 + 654*n + 360)*x^4 + (54*n^5 + 94*n^4 - 302*n^3 - 379*n^2 + 329*n + 204)*x^5 + (12*n^5 + 20*n^4 - 70*n^3 - 86*n^2 + 76*n + 48)*x^6)*v[i - 1] + ((3*n^4 + 33*n^3 + 108*n^2 + 123*n + 36)*x - (12*n^4 - 64*n^3 - 491*n^2 - 715*n - 228)*x^2 - (102*n^4 + 138*n^3 - 960*n^2 - 1878*n - 648)*x^3 - (228*n^4 + 532*n^3 - 1090*n^2 - 2822*n - 1032)*x^4 - (237*n^4 + 623*n^3 - 800*n^2 - 2503*n - 948)*x^5 - (120*n^4 + 324*n^3 - 363*n^2 - 1215*n - 468)*x^6 - (24*n^4 + 64*n^3 - 76*n^2 - 248*n - 96)*x^7)*deriv(v[i - 1], x) + ((6*n^3 + 28*n^2 + 37*n + 12)*x^2 + (42*n^3 + 196*n^2 + 259*n + 84)*x^3 + (120*n^3 + 560*n^2 + 740*n + 240)*x^4 + (180*n^3 + 840*n^2 + 1110*n + 360)*x^5 + (150*n^3 + 700*n^2 + 925*n + 300)*x^6 + (66*n^3 + 308*n^2 + 407*n + 132)*x^7 + (12*n^3 + 56*n^2 + 74*n + 24)*x^8)*deriv(deriv(v[i - 1], x), x))/(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (3*n^5 + 30*n^4 + 105*n^3 + 150*n^2 + 72*n)*x + (3*n^5 + 30*n^4 + 105*n^3 + 150*n^2 + 72*n)*x^2 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023