Real-rooted atlas

A304236 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-78\,n - 585\,n x + 324\,n x^{2} - 126\,n^{2} + 324\,n^{2} x + 81\,n^{3}\right)\,P_{n}(x) = \left(738 + 3780\,x + 2754\,x^{2} - 5832\,x^{3} - 141\,n - 4041\,n x - 8262\,n x^{2} + 5832\,n x^{3} - 1287\,n^{2} - 1458\,n^{2} x + 5832\,n^{2} x^{2} + 567\,n^{3} + 1458\,n^{3} x\right)\,P_{n-1}(x) + \left(65 + 102\,x - 2439\,x^{2} - 10368\,x^{3} - 11664\,x^{4} - 156\,n - 2304\,n x - 9396\,n x^{2} - 11664\,n x^{3} - 351\,n^{2} - 2025\,n^{2} x - 2916\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
213922
3251843
410024381424
52386842431165
6709304234027297882
71783901812474291626191
847833003360993422826561144652
9120858722821311118954032805534769
10310062588497822981009665492075590492632942
117769272763225500424021164259159535429410322419
12194857203814082639441609194614926275549155753144147538160
134833735564052254152085777179266272661330674403720087192294613
1411958251505801776909014202428720286584480204604785595213924782969851085202
Technical details
Node id
n-3ab28a4bfbab7260 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8301d56c1771be39

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:    13,     9;
  3:    25,    18;
  4:   100,   243,     81;
  5:   238,   684,    243;
  6:   709,  3042,   3402,    729;
  7:  1783,  9018,  12474,   2916;
  8:  4783, 30033,  60993,  42282,  6561;
  9: 12085, 87228, 213111, 189540, 32805;
%F Formula
The row polynomials satisfy (81*n^3-126*n^2-78*n+(324*n^2-585*n)*x+324*n*x^2)*P(n,x) = (567*n^3-1287*n^2-141*n+738+(1458*n^3-1458*n^2-4041*n+3780)*x+(5832*n^2-8262*n+2754)*x^2+(5832*n-5832)*x^3)*P(n-1,x) - (351*n^2+156*n-65+(2025*n^2+2304*n-102)*x+(2916*n^2+9396*n+2439)*x^2+(11664*n+10368)*x^3+11664*x^4)*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[((567*n^3 - 1287*n^2 - 141*n + 738 + (1458*n^3 - 1458*n^2 - 4041*n + 3780)*x + (5832*n^2 - 8262*n + 2754)*x^2 + (5832*n - 5832)*x^3)*P[n - 1, x] - (351*n^2 + 156*n - 65 + (2025*n^2 + 2304*n - 102)*x + (2916*n^2 + 9396*n + 2439)*x^2 + (11664*n + 10368)*x^3 + 11664*x^4)*D[P[n - 1, x], x])/(81*n^3 - 126*n^2 - 78*n + (324*n^2 - 585*n)*x + 324*n*x^2)]];
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] = ((567*n^3 - 1287*n^2 - 141*n + 738 + (1458*n^3 - 1458*n^2 - 4041*n + 3780)*x + (5832*n^2 - 8262*n + 2754)*x^2 + (5832*n - 5832)*x^3)*v[i - 1] - (351*n^2 + 156*n - 65 + (2025*n^2 + 2304*n - 102)*x + (2916*n^2 + 9396*n + 2439)*x^2 + (11664*n + 10368)*x^3 + 11664*x^4)*deriv(v[i - 1], x))/(81*n^3 - 126*n^2 - 78*n + (324*n^2 - 585*n)*x + 324*n*x^2)); 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.