Real-rooted atlas

A104978 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2} + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-2\,x - 12\,x^{2} - 18\,x^{3} - 8\,x^{4} - 7\,n - 20\,n x + 9\,n x^{2} + 50\,n x^{3} + 28\,n x^{4} + 13\,n^{2} + 46\,n^{2} x + 33\,n^{2} x^{2} - 20\,n^{2} x^{3} - 20\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(-1 - 9\,x - 29\,x^{2} - 43\,x^{3} - 30\,x^{4} - 8\,x^{5} + n + 15\,n x + 59\,n x^{2} + 97\,n x^{3} + 72\,n x^{4} + 20\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
2291021
35368566192
4141405168334982001
542540273068008361406622539
613220791342045540858308530534970267276
742980086306327209069569510554608806853120663287496

All 15 rows

Technical details
Node id
n-15250cda9c30cc03 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b24c47ab26a256b2

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:    2,      9,      10;
  3:    5,     36,      85,      66;
  4:   14,    140,     516,     833,      498;
  5:   42,    540,    2730,    6800,     8361,     4066;
  6:  132,   2079,   13420,   45540,    85830,    85305,     34970;
  7:  429,   8008,   63063,  272090,   695695,  1055460,    880685,    312066;
  8: 1430,  30888,  287560, 1509508,  4893420, 10042760,  12754952,   9173505,  2862562;
  9: 4862, 119340, 1283568, 7951104, 31298904, 81276048, 139351380, 152235072, 96220561, 26824386;
%F Formula
The row polynomials satisfy (4*n^2+2*n+(8*n^2+4*n)*x+(4*n^2+2*n)*x^2)*P(n,x) = (13*n^2-7*n+(46*n^2-20*n-2)*x+(33*n^2+9*n-12)*x^2-(20*n^2-50*n+18)*x^3-(20*n^2-28*n+8)*x^4)*P(n-1,x) + (n-1+(15*n-9)*x+(59*n-29)*x^2+(97*n-43)*x^3+(72*n-30)*x^4+(20*n-8)*x^5)*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[((13*n^2 - 7*n + (46*n^2 - 20*n - 2)*x + (33*n^2 + 9*n - 12)*x^2 - (20*n^2 - 50*n + 18)*x^3 - (20*n^2 - 28*n + 8)*x^4)*P[n - 1, x] + (n - 1 + (15*n - 9)*x + (59*n - 29)*x^2 + (97*n - 43)*x^3 + (72*n - 30)*x^4 + (20*n - 8)*x^5)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (8*n^2 + 4*n)*x + (4*n^2 + 2*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] = ((13*n^2 - 7*n + (46*n^2 - 20*n - 2)*x + (33*n^2 + 9*n - 12)*x^2 - (20*n^2 - 50*n + 18)*x^3 - (20*n^2 - 28*n + 8)*x^4)*v[i - 1] + (n - 1 + (15*n - 9)*x + (59*n - 29)*x^2 + (97*n - 43)*x^3 + (72*n - 30)*x^4 + (20*n - 8)*x^5)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (8*n^2 + 4*n)*x + (4*n^2 + 2*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

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.