Real-rooted atlas

A109692 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
28262155
348244412231935
4384267269568028346521505
538403401612032821244418718265835623645
6460804965122226304531687271323965095206151420521827575
764512081953284457408013454641624340908826390757615876943240883535894930575
8103219201511239689672404483534539136806548430411768202944107214457685575988952126738958542061737025
918579456030814986242269966694497475120128268887587232494119603632604873126864475618403016217972625850443586354752229272062325
10371589120068875960320574171785216283478941337691792814296322036900797468831368127078928331024200080162290857023462493881051176501729986783525131527051677175
1181749606400167505444864015593445568512870547538585603238401362270728428272769173121565966138648000207708270457025619273926055064161191606832677132441748229296500743894316915758549258359016375
12196199055360044043538268160452970252730368282220603432550411863736700802048354483410486878727719606934266566412345039342614214414388030964695601611918601644305272066607478638941204225475131375333003496303289504025606997343490162625
135101175439360012451935132057601402344702266572896478839781711872452389661971453952152670743900678912038149631963923887367146709765555644608100368581155541057601043673221951412753678102812832081600243983216752161867204124082705554130295017831146776470527546738795448742522125
14142832912302080037671064861409280461142159860957184347269710013454745617973246723194863616676292397006200627201907878786443602862084099308492371394295046741010643410754147208442556985018713878087927081480031054951525410955396369810830322538200789240591788207323820915218340995098071307270587901253879320022245629336375
Technical details
Node id
n-21cb97761b658085 · revision 0 · JSON
Definition source
derived:mobius_x_over_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:         2,          3;
  2:         8,         26,          21;
  3:        48,        244,         412,         231;
  4:       384,       2672,        6956,        8028,         3465;
  5:      3840,      34016,      120328,      212444,       187182,        65835;
  6:     46080,     496512,     2226304,     5316872,      7132396,      5095206,      1514205;
  7:    645120,    8195328,    44574080,   134546416,    243409088,    263907576,    158769432,     40883535;
  8:  10321920,  151123968,   967240448,  3534539136,   8065484304,  11768202944,  10721445768,   5575988952,   1267389585;
  9: 185794560, 3081498624, 22699666944, 97475120128, 268887587232, 494119603632, 604873126864, 475618403016, 217972625850, 44358635475;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n+(6*n-1)*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 + (6*n - 1)*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 + (6*n - 1)*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.