Real-rooted atlas

A166961 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 7\,x + 18\,x^{2} + 23\,x^{3} + 15\,x^{4} + 4\,x^{5} - 3\,n x - 15\,n x^{2} - 27\,n x^{3} - 21\,n x^{4} - 6\,n x^{5} + 2\,n^{2} x^{2} + 6\,n^{2} x^{3} + 6\,n^{2} x^{4} + 2\,n^{2} x^{5}\right)\,P_{n-1}(x) + \left(5\,x + 26\,x^{2} + 56\,x^{3} + 62\,x^{4} + 35\,x^{5} + 8\,x^{6} - 4\,n x^{2} - 16\,n x^{3} - 24\,n x^{4} - 16\,n x^{5} - 4\,n x^{6}\right)\,P_{n-1}'(x) + \left(2\,x^{2} + 10\,x^{3} + 20\,x^{4} + 20\,x^{5} + 10\,x^{6} + 2\,x^{7}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
21111123
3162163103329
413652014309114426913
51218025112712037629228023202811
61130673309251558100288690523594687128057861271
7178386459804134426295988853051353450288922427722857916385415249

All 15 rows

Technical details
Node id
n-bd1c359a0bd1db05 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-37e7a78a4ba18100

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,      11,        11;
  3: 1,      62,       163,         103;
  4: 1,     365,      2014,        3091,         1442;
  5: 1,    2180,     25112,       71203,        76292,        28023;
  6: 1,   13067,    330925,     1558100,      2886905,      2359468,       712805;
  7: 1,   78386,   4598041,    34426295,     98885305,    135345028,     89224277,     22857916;
  8: 1,  470297,  66227168,   786023693,   3318356125,   6789608513,   7338047150,   4044660409,    898132469;
  9: 1, 2821760, 973655134, 18713765511, 112811077687, 325026278933, 513069502844, 456837902539, 216006649223, 42280116147;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1-(3*n-7)*x+(2*n^2-15*n+18)*x^2+(6*n^2-27*n+23)*x^3+(6*n^2-21*n+15)*x^4+(2*n^2-6*n+4)*x^5)*P(n-1,x) + (5*x-(4*n-26)*x^2-(16*n-56)*x^3-(24*n-62)*x^4-(16*n-35)*x^5-(4*n-8)*x^6)*d/dx P(n-1,x) + (2*x^2+10*x^3+20*x^4+20*x^5+10*x^6+2*x^7)*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[((1 - (3*n - 7)*x + (2*n^2 - 15*n + 18)*x^2 + (6*n^2 - 27*n + 23)*x^3 + (6*n^2 - 21*n + 15)*x^4 + (2*n^2 - 6*n + 4)*x^5)*P[n - 1, x] + (5*x - (4*n - 26)*x^2 - (16*n - 56)*x^3 - (24*n - 62)*x^4 - (16*n - 35)*x^5 - (4*n - 8)*x^6)*D[P[n - 1, x], x] + (2*x^2 + 10*x^3 + 20*x^4 + 20*x^5 + 10*x^6 + 2*x^7)*D[P[n - 1, x], {x, 2}])/(1 + 2*x + 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] = ((1 - (3*n - 7)*x + (2*n^2 - 15*n + 18)*x^2 + (6*n^2 - 27*n + 23)*x^3 + (6*n^2 - 21*n + 15)*x^4 + (2*n^2 - 6*n + 4)*x^5)*v[i - 1] + (5*x - (4*n - 26)*x^2 - (16*n - 56)*x^3 - (24*n - 62)*x^4 - (16*n - 35)*x^5 - (4*n - 8)*x^6)*deriv(v[i - 1], x) + (2*x^2 + 10*x^3 + 20*x^4 + 20*x^5 + 10*x^6 + 2*x^7)*deriv(deriv(v[i - 1], x), x))/(1 + 2*x + 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. [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.