Real-rooted atlas

A181543 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n^{2} + 3\,n^{2} x + 3\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n}(x) = \left(12\,x + 72\,x^{2} + 168\,x^{3} + 192\,x^{4} + 108\,x^{5} + 24\,x^{6} - 18\,n x - 108\,n x^{2} - 252\,n x^{3} - 288\,n x^{4} - 162\,n x^{5} - 36\,n x^{6} + n^{2} + 11\,n^{2} x + 45\,n^{2} x^{2} + 91\,n^{2} x^{3} + 98\,n^{2} x^{4} + 54\,n^{2} x^{5} + 12\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(6\,x + 72\,x^{2} + 276\,x^{3} + 504\,x^{4} + 486\,x^{5} + 240\,x^{6} + 48\,x^{7} + 3\,n x - 12\,n x^{2} - 102\,n x^{3} - 228\,n x^{4} - 237\,n x^{5} - 120\,n x^{6} - 24\,n x^{7}\right)\,P_{n-1}'(x) + \left(6\,x^{2} + 42\,x^{3} + 120\,x^{4} + 180\,x^{5} + 150\,x^{6} + 66\,x^{7} + 12\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
21101021
31308456171
41684146923461521
51130151047605630225214283
61222447023680498004555215184138909
7135011340943603138804929123673601049601385163

All 15 rows

Technical details
Node id
n-7f5c7b0cb55ac048 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e0dcd909a84c02c1

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,   2;
  2: 1,  10,    10;
  3: 1,  30,    84,     56;
  4: 1,  68,   414,    692,     346;
  5: 1, 130,  1510,   4760,    5630,     2252;
  6: 1, 222,  4470,  23680,   49800,    45552,    15184;
  7: 1, 350, 11340,  94360,  313880,   492912,   367360,   104960;
  8: 1, 520, 25564, 318136, 1568350,  3760792,  4703020,  2956648,   739162;
  9: 1, 738, 52524, 939792, 6577326, 22576932, 42125916, 43721568, 23764194, 5280932;
%F Formula
The row polynomials satisfy (n^2+3*n^2*x+3*n^2*x^2+n^2*x^3)*P(n,x) = (n^2+(11*n^2-18*n+12)*x+(45*n^2-108*n+72)*x^2+(91*n^2-252*n+168)*x^3+(98*n^2-288*n+192)*x^4+(54*n^2-162*n+108)*x^5+(12*n^2-36*n+24)*x^6)*P(n-1,x) + ((3*n+6)*x-(12*n-72)*x^2-(102*n-276)*x^3-(228*n-504)*x^4-(237*n-486)*x^5-(120*n-240)*x^6-(24*n-48)*x^7)*d/dx P(n-1,x) + (6*x^2+42*x^3+120*x^4+180*x^5+150*x^6+66*x^7+12*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^2 + (11*n^2 - 18*n + 12)*x + (45*n^2 - 108*n + 72)*x^2 + (91*n^2 - 252*n + 168)*x^3 + (98*n^2 - 288*n + 192)*x^4 + (54*n^2 - 162*n + 108)*x^5 + (12*n^2 - 36*n + 24)*x^6)*P[n - 1, x] + ((3*n + 6)*x - (12*n - 72)*x^2 - (102*n - 276)*x^3 - (228*n - 504)*x^4 - (237*n - 486)*x^5 - (120*n - 240)*x^6 - (24*n - 48)*x^7)*D[P[n - 1, x], x] + (6*x^2 + 42*x^3 + 120*x^4 + 180*x^5 + 150*x^6 + 66*x^7 + 12*x^8)*D[P[n - 1, x], {x, 2}])/(n^2 + 3*n^2*x + 3*n^2*x^2 + n^2*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^2 + (11*n^2 - 18*n + 12)*x + (45*n^2 - 108*n + 72)*x^2 + (91*n^2 - 252*n + 168)*x^3 + (98*n^2 - 288*n + 192)*x^4 + (54*n^2 - 162*n + 108)*x^5 + (12*n^2 - 36*n + 24)*x^6)*v[i - 1] + ((3*n + 6)*x - (12*n - 72)*x^2 - (102*n - 276)*x^3 - (228*n - 504)*x^4 - (237*n - 486)*x^5 - (120*n - 240)*x^6 - (24*n - 48)*x^7)*deriv(v[i - 1], x) + (6*x^2 + 42*x^3 + 120*x^4 + 180*x^5 + 150*x^6 + 66*x^7 + 12*x^8)*deriv(deriv(v[i - 1], x), x))/(n^2 + 3*n^2*x + 3*n^2*x^2 + n^2*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

Same fingerprint ?

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