Real-rooted atlas

A128813 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
28364084
3564049647602184
4616618023128382442356091728
598561272166543281675792213618410837605687136
62168323429536225370407874441615424673616055849669360640489093696
762881281178872649450847204199236464111664307041776716894415658930720589565440055756681344

All 15 rows

Technical details
Node id
n-bee118c2057fa6b7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4aae017df98b90e3

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:           2,            4;
  2:           8,           36,            40;
  3:          56,          404,           964,            760;
  4:         616,         6180,         23128,          38244,          23560;
  5:        9856,       127216,        654328,        1675792,        2136184,         1083760;
  6:      216832,      3429536,      22537040,       78744416,      154246736,       160558496,        69360640;
  7:     6288128,    117887264,     945084720,     4199236464,    11166430704,     17767168944,     15658930720,      5895654400;
  8:   232660736,   5047234720,   47817846416,   258385983648,   870874710624,   1874526197664,   2516001851536,   1924962661280,    642626329600;
  9: 10702393856, 263814657216, 2886044857056, 18388982360384, 75200730464832, 204667165737408, 370671648052960, 430724534227776, 291355733095680, 87397180825600;
%F Formula
The row polynomials satisfy (2+2*x)*P(n,x) = (n^2+n+2+(4*n^2+4*n+4)*x+(3*n^2+3*n+2)*x^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 + n + 2 + (4*n^2 + 4*n + 4)*x + (3*n^2 + 3*n + 2)*x^2)*P[n - 1, x])/(2 + 2*x)]];
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 + n + 2 + (4*n^2 + 4*n + 4)*x + (3*n^2 + 3*n + 2)*x^2)*v[i - 1])/(2 + 2*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.