Real-rooted atlas

A143497 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-5\,x - 11\,x^{2} - 6\,x^{3} - 4\,n - 13\,n x - 13\,n x^{2} - 3\,n x^{3} + n x^{4} + n^{2} x + 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}(x) + \left(4 + 18\,x + 30\,x^{2} + 22\,x^{3} + 6\,x^{4} - 2\,n x - 8\,n x^{2} - 12\,n x^{3} - 8\,n x^{4} - 2\,n x^{5}\right)\,P_{n-1}'(x) + \left(x + 5\,x^{2} + 10\,x^{3} + 10\,x^{4} + 5\,x^{5} + x^{6}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 20 + 50\,x + 31\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0205031101
11204505582291357
2840420078126412196121225
36720420001041601282407844019081378641
46048045360014061602308320211788010303742077757584589
5604800529200019686240403956004941720036063090145442502501801168504981
666528006652800028873152071096256010871784001057850640639947000220158488329899694110999377
779833600898128000445471488012797326080234830534402856196728023038092000118885583523562916652472630861109237221145

All 15 rows

Technical details
Node id
n-562ff9e6276371c2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e38b424e1840fcda

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:          20,           50,            31;
  1:         120,          450,           558,           229;
  2:         840,         4200,          7812,          6412,           1961;
  3:        6720,        42000,        104160,        128240,          78440,          19081;
  4:       60480,       453600,       1406160,       2308320,        2117880,        1030374,         207775;
  5:      604800,      5292000,      19686240,      40395600,       49417200,       36063090,       14544250,        2501801;
  6:     6652800,     66528000,     288731520,     710962560,     1087178400,     1057850640,      639947000,      220158488,       32989969;
  7:    79833600,    898128000,    4454714880,   12797326080,    23483053440,    28561967280,    23038092000,    11888558352,     3562916652,     472630861;
  8:  1037836800,  12972960000,   72389116800,  237664627200,   508799491200,   742611149280,   748737990000,   515170861920,   231589582380,   61442011930,    7307593151;
  9: 14529715200, 199783584000, 1238658220800, 4575044073600, 11193588806400, 19060352831520, 23061130092000, 19834078183920, 11888265228840, 4731034918610, 1125369345254, 121247816845;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(4*n-(n^2-13*n-5)*x-(3*n^2-13*n-11)*x^2-(3*n^2-3*n-6)*x^3-(n^2+n)*x^4)*P(n-1,x) + (4-(2*n-18)*x-(8*n-30)*x^2-(12*n-22)*x^3-(8*n-6)*x^4-2*n*x^5)*d/dx P(n-1,x) + (x+5*x^2+10*x^3+10*x^4+5*x^5+x^6)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 20 + 50*x + 31*x^2.
%t Mathematica
P[0, x_] := 20 + 50*x + 31*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - (n^2 - 13*n - 5)*x - (3*n^2 - 13*n - 11)*x^2 - (3*n^2 - 3*n - 6)*x^3 - (n^2 + n)*x^4)*P[n - 1, x] + (4 - (2*n - 18)*x - (8*n - 30)*x^2 - (12*n - 22)*x^3 - (8*n - 6)*x^4 - 2*n*x^5)*D[P[n - 1, x], x] + (x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6)*D[P[n - 1, x], {x, 2}])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 20 + 50*x + 31*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(4*n - (n^2 - 13*n - 5)*x - (3*n^2 - 13*n - 11)*x^2 - (3*n^2 - 3*n - 6)*x^3 - (n^2 + n)*x^4)*v[i - 1] + (4 - (2*n - 18)*x - (8*n - 30)*x^2 - (12*n - 22)*x^3 - (8*n - 6)*x^4 - 2*n*x^5)*deriv(v[i - 1], x) + (x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6)*deriv(deriv(v[i - 1], x), x))/(1 + x)); 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. [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.