Real-rooted atlas

A121574 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(17\,n + 11\,n x + 7\,n x^{3} + n x^{4} + n^{2} + 2\,n^{2} x + 8\,n^{2} x^{2} + n^{2} x^{3} - 3\,n^{3} + 3\,n^{3} x\right)\,P_{n}(x) = \left(3 + 49\,x + 102\,x^{2} + 70\,x^{3} + 15\,x^{4} + x^{5} + 106\,n + 98\,n x - 19\,n x^{2} - 26\,n x^{3} - 3\,n x^{4} - 25\,n^{2} - 56\,n^{2} x - 35\,n^{2} x^{2} - 4\,n^{2} x^{3} - 24\,n^{3} - 12\,n^{3} x\right)\,P_{n-1}(x) + \left(104 + 405\,x + 609\,x^{2} + 438\,x^{3} + 150\,x^{4} + 21\,x^{5} + x^{6} + 143\,n + 370\,n x + 341\,n x^{2} + 133\,n x^{3} + 20\,n x^{4} + n x^{5} + 39\,n^{2} + 69\,n^{2} x + 33\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1415
2179127
36857141140
4264307121191712
5100215008282092413564
6373968704917174332129117620
713768300432653812235316445734186240

All 15 rows

Technical details
Node id
n-9c3fe92d57e53218 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8325caf9cc6f8183

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,      1;
  2:     17,      9,      1;
  3:     68,     57,     14,      1;
  4:    264,    307,    121,     19,      1;
  5:   1002,   1500,    828,    209,     24,     1;
  6:   3739,   6870,   4917,   1743,    321,    29,    1;
  7:  13768,  30043,  26538,  12235,   3164,   457,   34,   1;
  8:  50159, 126897, 133705,  76301,  25686,  5203,  617,  39,  1;
  9: 181142, 521691, 639548, 436477, 182118, 48014, 7972, 801, 44, 1;
%F Formula
The row polynomials satisfy -(3*n^3-n^2-17*n-(3*n^3+2*n^2+11*n)*x-8*n^2*x^2-(n^2+7*n)*x^3-n*x^4)*P(n,x) = -(24*n^3+25*n^2-106*n-3+(12*n^3+56*n^2-98*n-49)*x+(35*n^2+19*n-102)*x^2+(4*n^2+26*n-70)*x^3+(3*n-15)*x^4-x^5)*P(n-1,x) + (39*n^2+143*n+104+(69*n^2+370*n+405)*x+(33*n^2+341*n+609)*x^2+(3*n^2+133*n+438)*x^3+(20*n+150)*x^4+(n+21)*x^5+x^6)*d/dx 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[(-(24*n^3 + 25*n^2 - 106*n - 3 + (12*n^3 + 56*n^2 - 98*n - 49)*x + (35*n^2 + 19*n - 102)*x^2 + (4*n^2 + 26*n - 70)*x^3 + (3*n - 15)*x^4 - x^5)*P[n - 1, x] + (39*n^2 + 143*n + 104 + (69*n^2 + 370*n + 405)*x + (33*n^2 + 341*n + 609)*x^2 + (3*n^2 + 133*n + 438)*x^3 + (20*n + 150)*x^4 + (n + 21)*x^5 + x^6)*D[P[n - 1, x], x])/(-3*n^3 + n^2 + 17*n + (3*n^3 + 2*n^2 + 11*n)*x + 8*n^2*x^2 + (n^2 + 7*n)*x^3 + n*x^4)]];
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] = (-(24*n^3 + 25*n^2 - 106*n - 3 + (12*n^3 + 56*n^2 - 98*n - 49)*x + (35*n^2 + 19*n - 102)*x^2 + (4*n^2 + 26*n - 70)*x^3 + (3*n - 15)*x^4 - x^5)*v[i - 1] + (39*n^2 + 143*n + 104 + (69*n^2 + 370*n + 405)*x + (33*n^2 + 341*n + 609)*x^2 + (3*n^2 + 133*n + 438)*x^3 + (20*n + 150)*x^4 + (n + 21)*x^5 + x^6)*deriv(v[i - 1], x))/(-3*n^3 + n^2 + 17*n + (3*n^3 + 2*n^2 + 11*n)*x + 8*n^2*x^2 + (n^2 + 7*n)*x^3 + n*x^4)); 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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.