Real-rooted atlas

A088996 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(-1 - 2\,x - x^{2} + 2\,n + 6\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
23162140
315113281231640
4105101636625832346514080
5945111395226212206614199365835394240
610395144264831079254475243694413990024151420513404160
71351352156097146991555552090912551103716984521912741531340883535536166400
820270253653064028732633212884874403603813734643852043271764314844563127728126738958524663654400
934459425691966755616312004431960708100106361893878235588328034347347550348328748273316181255093425443586354751282510028800
1065472907514491285920144085984281847615135616326734359958286242920938881578754824993819792771656576162650294343997911762717600172998678352574385581670400
111374931057533247035454536489309644612399561517201910506166642271032160590875367470238307328588210945127795354301192654799355147865543282748757376535519310825743894316915754760677226905600
123162341432258293035751560995515188462187233989042861043544212240807223123348342232018723127028839699796458185798373751344788731610233335917596227107291332049340851233340554194081263365150003496303289504025333247405883392000
137905853580625223453835093475291173279448501023162100068309718125498650138771879489125679861215173141083375530824457230494296675906797644939304742644658319592158798901284614151075971055520931703001586571313243254107722202539986562517831146776470527525326802847137792000
14213458046676875646807549445820090906746381236395785522005541037936466237905750387522320108819113885263016649944237856894379591599304575669018050883010798597688931616334315446365488030533524635924612456130533453619606782308226922081941722561080247508456406162102663965180098071307270587901252076797833465298944000
Technical details
Node id
n-3022f3f38e90eea2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a46d1ee90c8acbaf

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:        1,         3;
  2:        3,        16,         21;
  3:       15,       113,        281,         231;
  4:      105,      1016,       3662,        5832,         3465;
  5:      945,     11139,      52262,      122066,       141993,        65835;
  6:    10395,    144264,     831079,     2544752,      4369441,      3990024,      1514205;
  7:   135135,   2156097,   14699155,    55520909,    125511037,    169845219,    127415313,     40883535;
  8:  2027025,  36530640,  287326332,  1288487440,   3603813734,   6438520432,   7176431484,   4563127728,   1267389585;
  9: 34459425, 691966755, 6163120044, 31960708100, 106361893878, 235588328034, 347347550348, 328748273316, 181255093425, 44358635475;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n-1+(6*n-2)*x+(4*n-1)*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[((2*n - 1 + (6*n - 2)*x + (4*n - 1)*x^2)*P[n - 1, x])/(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] = ((2*n - 1 + (6*n - 2)*x + (4*n - 1)*x^2)*v[i - 1])/(1 + 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

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.