Real-rooted atlas

A371259 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(25 + 87\,x + 112\,x^{2} + 65\,x^{3} + 17\,x^{4} + 2\,x^{5} + 10\,n + 21\,n x - 26\,n x^{3} - 18\,n x^{4} - 3\,n x^{5} + n^{2} + n^{2} x - 2\,n^{2} x^{2} - 2\,n^{2} x^{3} + n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}(x) + \left(11\,x + 48\,x^{2} + 82\,x^{3} + 68\,x^{4} + 27\,x^{5} + 4\,x^{6} + 2\,n x + 6\,n x^{2} + 4\,n x^{3} - 4\,n x^{4} - 6\,n x^{5} - 2\,n x^{6}\right)\,P_{n-1}'(x) + \left(x^{2} + 5\,x^{3} + 10\,x^{4} + 10\,x^{5} + 5\,x^{6} + x^{7}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1363773
21764362818657257
3112896348552358616122961943025
4914457637677600581994083994466410278281155244529
591445760047144473929719218080100156380885159124044106271424531585599449
61106493696006852816276481767824516160243149325587218806086168367755236108521332159917777784596988745
715933509222400115259348224512357200537783040614791862288640634676291813312392996550284960135150510160736199130881813772285921697958977

All 15 rows

Technical details
Node id
n-10b0c04244f0dc1a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-7819055e160ae812

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:                 36,                  37;
  2:               1764,                3628,                 1865;
  3:             112896,              348552,               358616,               122961;
  4:            9144576,            37677600,             58199408,             39944664,             10278281;
  5:          914457600,          4714447392,           9719218080,          10015638088,           5159124044,           1062714245;
  6:       110649369600,        685281627648,        1767824516160,        2431493255872,        1880608616836,         775523610852,         133215991777;
  7:     15933509222400,     115259348224512,      357200537783040,      614791862288640,      634676291813312,      392996550284960,      135150510160736,       19913088181377;
  8:   2692763058585600,   22288095576778752,    80678860154654976,   166819816171498752,   215506138966152128,   178114983558813760,    91976017747496448,    27131156184831280,    3500271565033393;
  9: 527781559482777600, 4920574797724723200, 20380454761471617024, 49221357628032294912, 76390319392998317568, 79007389959798036736, 54456139494941236032, 24120500031527459568, 6230071236777009652, 714944028588443461;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (n^2+10*n+25+(n^2+21*n+87)*x-(2*n^2-112)*x^2-(2*n^2+26*n-65)*x^3+(n^2-18*n+17)*x^4+(n^2-3*n+2)*x^5)*P(n-1,x) + ((2*n+11)*x+(6*n+48)*x^2+(4*n+82)*x^3-(4*n-68)*x^4-(6*n-27)*x^5-(2*n-4)*x^6)*d/dx P(n-1,x) + (x^2+5*x^3+10*x^4+10*x^5+5*x^6+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[((n^2 + 10*n + 25 + (n^2 + 21*n + 87)*x - (2*n^2 - 112)*x^2 - (2*n^2 + 26*n - 65)*x^3 + (n^2 - 18*n + 17)*x^4 + (n^2 - 3*n + 2)*x^5)*P[n - 1, x] + ((2*n + 11)*x + (6*n + 48)*x^2 + (4*n + 82)*x^3 - (4*n - 68)*x^4 - (6*n - 27)*x^5 - (2*n - 4)*x^6)*D[P[n - 1, x], x] + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + 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] = ((n^2 + 10*n + 25 + (n^2 + 21*n + 87)*x - (2*n^2 - 112)*x^2 - (2*n^2 + 26*n - 65)*x^3 + (n^2 - 18*n + 17)*x^4 + (n^2 - 3*n + 2)*x^5)*v[i - 1] + ((2*n + 11)*x + (6*n + 48)*x^2 + (4*n + 82)*x^3 - (4*n - 68)*x^4 - (6*n - 27)*x^5 - (2*n - 4)*x^6)*deriv(v[i - 1], x) + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + 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.