Real-rooted atlas

A122542 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(10 - 29\,x + 43\,x^{2} - 5\,x^{3} - 3\,x^{4} - 19\,n + 44\,n x - 32\,n x^{2} - 2\,n x^{3} + n x^{4} + 11\,n^{2} - 17\,n^{2} x + 5\,n^{2} x^{2} + n^{2} x^{3} - 2\,n^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(6 + 23\,x - 61\,x^{2} + 11\,x^{3} + 5\,x^{4} - 13\,n - 50\,n x + 60\,n x^{2} - 2\,n x^{3} - 3\,n x^{4} + 9\,n^{2} + 33\,n^{2} x - 15\,n^{2} x^{2} - 3\,n^{2} x^{3} - 2\,n^{3} - 6\,n^{3} x\right)\,P_{n-1}(x) + \left(3\,x + 10\,x^{2} - 41\,x^{3} + 17\,x^{4} + 10\,x^{5} + x^{6} - 5\,n x - 24\,n x^{2} + 32\,n x^{3} + 12\,n x^{4} + n x^{5} + 2\,n^{2} x + 12\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
111
211
3325
458215
572214245
694454202129
71174146102262361
813112314352166322991
9151585829387022463822681
10172129742092225012363424427169
11192741514411059224650199445450218989
12213442226735213486141848634301658256249903
1323422313412242275263695830046147944342726622130277
142550842621926851602852768849457956238186012886682338177
1527602563428982904101787702287061908621038663649080661062742873553
Technical details
Node id
n-be4af3be90cb301e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8cc5447af65adb41

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  1;
   3:  3,   2;
   4:  5,   8,   2;
   5:  7,  22,  14,    2;
   6:  9,  44,  54,   20,    2;
   7: 11,  74, 146,  102,   26,    2;
   8: 13, 112, 314,  352,  166,   32,   2;
   9: 15, 158, 582,  938,  702,  246,  38,  2;
  10: 17, 212, 974, 2092, 2250, 1236, 342, 44, 2;
%F Formula
The row polynomials satisfy -(2*n^3-11*n^2+19*n-10-(2*n^3-17*n^2+44*n-29)*x-(5*n^2-32*n+43)*x^2-(n^2-2*n-5)*x^3-(n-3)*x^4)*P(n,x) = -(2*n^3-9*n^2+13*n-6+(6*n^3-33*n^2+50*n-23)*x+(15*n^2-60*n+61)*x^2+(3*n^2+2*n-11)*x^3+(3*n-5)*x^4)*P(n-1,x) + ((2*n^2-5*n+3)*x+(12*n^2-24*n+10)*x^2+(2*n^2+32*n-41)*x^3+(12*n+17)*x^4+(n+10)*x^5+x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 9*n^2 + 13*n - 6 + (6*n^3 - 33*n^2 + 50*n - 23)*x + (15*n^2 - 60*n + 61)*x^2 + (3*n^2 + 2*n - 11)*x^3 + (3*n - 5)*x^4)*P[n - 1, x] + ((2*n^2 - 5*n + 3)*x + (12*n^2 - 24*n + 10)*x^2 + (2*n^2 + 32*n - 41)*x^3 + (12*n + 17)*x^4 + (n + 10)*x^5 + x^6)*D[P[n - 1, x], x])/(-2*n^3 + 11*n^2 - 19*n + 10 + (2*n^3 - 17*n^2 + 44*n - 29)*x + (5*n^2 - 32*n + 43)*x^2 + (n^2 - 2*n - 5)*x^3 + (n - 3)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n^3 - 9*n^2 + 13*n - 6 + (6*n^3 - 33*n^2 + 50*n - 23)*x + (15*n^2 - 60*n + 61)*x^2 + (3*n^2 + 2*n - 11)*x^3 + (3*n - 5)*x^4)*v[i - 1] + ((2*n^2 - 5*n + 3)*x + (12*n^2 - 24*n + 10)*x^2 + (2*n^2 + 32*n - 41)*x^3 + (12*n + 17)*x^4 + (n + 10)*x^5 + x^6)*deriv(v[i - 1], x))/(-2*n^3 + 11*n^2 - 19*n + 10 + (2*n^3 - 17*n^2 + 44*n - 29)*x + (5*n^2 - 32*n + 43)*x^2 + (n^2 - 2*n - 5)*x^3 + (n - 3)*x^4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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.