Real-rooted atlas

A089231 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 3\,x + 2\,x^{2} - x^{3} - x^{4} + 2\,n x + 5\,n x^{2} + 4\,n x^{3} + n x^{4}\right)\,P_{n-1}(x) + \left(-x^{2} - 3\,x^{3} - 3\,x^{4} - x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
2181322
31156373152
41241865445011256
513543022905225405112032
61488557160294155565637633130768
71631533185151187554007016530233943531586944
8180254841888385630201857658177008379008459655321241984
919939968576410711268012466356202849006483611678004958941091310579712
1011205985162480264285026736192169175370656418960148454860517572114008240731414920975104
111143863528924559403307808770266628392636660812101266395344525994697355283941299511247016223383939634176
1211681207848928012381435205148592226962027616794002016819848348552559462307604823011479664903715062082029764012131532861774848
13119516458793078242449354943251176881940780658920679564314825968311897628103085541727264106694573297139109013522796473353501752440329825546764288
141224219311239784450460011108379272189639480032281313068321929281677323113541150442724552724768692111997602230650419549727557005117567956449226465573803186921615785559906304
Technical details
Node id
n-0c674e242339be8f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-747352d83647270b

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,  3;
  2: 1,  8,   13;
  3: 1, 15,   63,    73;
  4: 1, 24,  186,   544,     501;
  5: 1, 35,  430,  2290,    5225,    4051;
  6: 1, 48,  855,  7160,   29415,   55656,    37633;
  7: 1, 63, 1533, 18515,  118755,  400701,   653023,   394353;
  8: 1, 80, 2548, 41888,  385630, 2018576,  5817700,  8379008,   4596553;
  9: 1, 99, 3996, 85764, 1071126, 8012466, 35620284, 90064836, 116780049, 58941091;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(2*n+3)*x+(5*n+2)*x^2+(4*n-1)*x^3+(n-1)*x^4)*P(n-1,x) - (x^2+3*x^3+3*x^4+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[((1 + (2*n + 3)*x + (5*n + 2)*x^2 + (4*n - 1)*x^3 + (n - 1)*x^4)*P[n - 1, x] - (x^2 + 3*x^3 + 3*x^4 + x^5)*D[P[n - 1, x], x])/(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 + (2*n + 3)*x + (5*n + 2)*x^2 + (4*n - 1)*x^3 + (n - 1)*x^4)*v[i - 1] - (x^2 + 3*x^3 + 3*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 - 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. [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.