Real-rooted atlas

A143494 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-2 - 13\,x - 34\,x^{2} - 43\,x^{3} - 26\,x^{4} - 6\,x^{5} + n x + 5\,n x^{2} + 9\,n x^{3} + 7\,n x^{4} + 2\,n x^{5}\right)\,P_{n-2}(x) + \left(-x - 6\,x^{2} - 14\,x^{3} - 16\,x^{4} - 9\,x^{5} - 2\,x^{6}\right)\,P_{n-2}'(x) + \left(3 + 13\,x + 19\,x^{2} + 11\,x^{3} + 2\,x^{4} - n x - 3\,n x^{2} - 3\,n x^{3} - n x^{4}\right)\,P_{n-1}(x) + \left(x + 4\,x^{2} + 6\,x^{3} + 4\,x^{4} + x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2246
34181739
486212777274
5161946518403722073
632582286359175513191516822
76417141166334432507233677810481145855
8128501445603180889375226426456252208608141346338
925614658174443895472245673038948743587965178732037293913184657
1051242950660103427794914929519309886193955476330545405131198532409837136529510
1110241262902487783200093848652190922597620037496352439920116426481006399832040163607861490290167
1220483727269377707924810174862902641555492644322337106144376459734042504266234654605078738625711639320517097863218
13409611038423544957142495165626800159241029647560425925691806443299121845200787521941327873716212991624706433431524865549453205627497065
1481923278758134598735194929850914591808645663606663871990311859104111439293545965701314286085663677104282637760681982656443845441074126667130651562586004500502
1516384976254651375100789511917927890239744742002507545414782488402273615706152328631065625048679354711741867150044192593451028549928818937838772744758908302625419036045333932679371727
Technical details
Node id
n-69f86626fdfebb46 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-21399122fd279ad0

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 -1)
  • 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:   2,     4;
   3:   4,    18,     17;
   4:   8,    62,    127,      77;
   5:  16,   194,    651,     840,      372;
   6:  32,   582,   2863,    5917,     5513,     1915;
   7:  64,  1714,  11663,   34432,    50723,    36778,    10481;
   8: 128,  5014,  45603,  180889,   375226,   426456,   252208,    60814;
   9: 256, 14658, 174443,  895472,  2456730,  3894874,  3587965,  1787320,   372939;
  10: 512, 42950, 660103, 4277949, 14929519, 30988619, 39554763, 30545405, 13119853, 2409837;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(2-(n-13)*x-(5*n-34)*x^2-(9*n-43)*x^3-(7*n-26)*x^4-(2*n-6)*x^5)*P(n-2,x) - (x+6*x^2+14*x^3+16*x^4+9*x^5+2*x^6)*d/dx P(n-2,x) + (3-(n-13)*x-(3*n-19)*x^2-(3*n-11)*x^3-(n-2)*x^4)*P(n-1,x) + (x+4*x^2+6*x^3+4*x^4+x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2 - (n - 13)*x - (5*n - 34)*x^2 - (9*n - 43)*x^3 - (7*n - 26)*x^4 - (2*n - 6)*x^5)*P[n - 2, x] - (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*D[P[n - 2, x], x] + (3 - (n - 13)*x - (3*n - 19)*x^2 - (3*n - 11)*x^3 - (n - 2)*x^4)*P[n - 1, x] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = (-(2 - (n - 13)*x - (5*n - 34)*x^2 - (9*n - 43)*x^3 - (7*n - 26)*x^4 - (2*n - 6)*x^5)*v[i - 2] - (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*deriv(v[i - 2], x) + (3 - (n - 13)*x - (3*n - 19)*x^2 - (3*n - 11)*x^3 - (n - 2)*x^4)*v[i - 1] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*deriv(v[i - 1], 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, v[i])))};
T(10)

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. [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
  4. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.