Real-rooted atlas

A109692 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-3 + 6\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
22730865
3405612300481365
485051609211196340838439585
52296355195344632122039764444838401464645
6757795519900242215204041228975239144966600964608065909025
7295540245882200808111789914478058598432472187280546688110384646451203493178325

All 15 rows

Technical details
Node id
n-58093636829549bd · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-6f5c8093b03ddefa

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:           27,            30,             8;
  3:          405,           612,           300,            48;
  4:         8505,         16092,         11196,          3408,           384;
  5:       229635,        519534,        463212,        203976,         44448,         3840;
  6:      7577955,      19900242,      21520404,      12289752,       3914496,       660096,        46080;
  7:    295540245,     882200808,    1117899144,     780585984,     324721872,     80546688,     11038464,      645120;
  8:  13299311025,   44427680280,   64420674408,   53012755584,   27101859984,   8820150912,   1785477888,   205645824,   10321920;
  9: 678264862275, 2505199292730, 4085152639848, 3863222674128, 2336424459696, 937661176224, 249822088704, 42626539008, 4228042752, 185794560;
%F Formula
The row polynomials satisfy P(n,x) = (6*n-3+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[(6*n - 3 + 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] = (6*n - 3 + 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.