Real-rooted atlas

A088996 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
22126855
323141224448935
4346580286956267238421505
565835187182212444120328340163840623645
6151420550952067132396531687222263044965124608021827575
740883535158769432263907576243409088134546416445740808195328645120894930575

All 15 rows

Technical details
Node id
n-45bcdc2be26ae0cd · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-2b67a3b9cedf1278

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:           3,            2;
  2:          21,           26,            8;
  3:         231,          412,          244,           48;
  4:        3465,         8028,         6956,         2672,          384;
  5:       65835,       187182,       212444,       120328,        34016,         3840;
  6:     1514205,      5095206,      7132396,      5316872,      2226304,       496512,       46080;
  7:    40883535,    158769432,    263907576,    243409088,    134546416,     44574080,     8195328,      645120;
  8:  1267389585,   5575988952,  10721445768,  11768202944,   8065484304,   3534539136,   967240448,   151123968,   10321920;
  9: 44358635475, 217972625850, 475618403016, 604873126864, 494119603632, 268887587232, 97475120128, 22699666944, 3081498624, 185794560;
%F Formula
The row polynomials satisfy P(n,x) = (4*n-1+2*n*x)*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[(4*n - 1 + 2*n*x)*P[n - 1, 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] = (4*n - 1 + 2*n*x)*v[i - 1]); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.