Real-rooted atlas

A109692 → basis A007318 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
221321265
32315624521201365
4346511664146488128168039585
565835283986488264418096178224302401464645
615142057980048174777642035801613297264461644866528065909025
7408835352548306266793808761004088296888334544470372960137990208172972803493178325
81267389585912625545628705725936515081634565766101974441231598080183888852484675921920518918400213083877825
944358635475362510186850131499309326427787804027843769413248544340358060409620454853184007888793656321771434892801764322560014702787569925
10172998678352515823525435200650601177375961583421732526082526007719990082759773470044162091099903732481084947373588483688601197593674195383910406704425728001132114642884225
11743894316915757530710386216503462173130995028954123839484117617512204472566880224762583451482242058277816023513613447893302106880614287748403686418682526538040323404496430540802815858805760096229744645159125
1234963032895040253881625267303000019736340493336221660769816858330656012619705763733374561861945547960043008200129845740786969615788587805698396169073183336466490883703802389944852481019407552985272321698413721919488012952950506496008949366251999798625
1317831146776470527521544440507997312501200634628525297301640860776844416745360947454078242055382561580577517646290662081951634987258035048961805867206794553052161252161740444710842886425530887105120204823717990469949151232596322876310530048091526690854287360064764752532480000903885991451979661125
1498071307270587901251281232420532793036007766890244320990033802895685425846581537664741747937992980885352013809428369561697707264192691110252091623445122047109856856343105126416638572489136496117504102957153863092546641924774276154883968228352160874906734804569292837235403317754427392052986474450601574400349729663675392000098523573068265783062625
Technical details
Node id
n-6e91a3cc96dd01b8 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-48fd4cbb64e3687d

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,           32,            12;
  3:         231,          562,           452,           120;
  4:        3465,        11664,         14648,          8128,          1680;
  5:       65835,       283986,        488264,        418096,        178224,         30240;
  6:     1514205,      7980048,      17477764,      20358016,      13297264,       4616448,        665280;
  7:    40883535,    254830626,     679380876,    1004088296,     888334544,     470372960,     137990208,     17297280;
  8:  1267389585,   9126255456,   28705725936,   51508163456,   57661019744,   41231598080,   18388885248,   4675921920,    518918400;
  9: 44358635475, 362510186850, 1314993093264, 2778780402784, 3769413248544, 3403580604096, 2045485318400, 788879365632, 177143489280, 17643225600;
%F Formula
The row polynomials satisfy P(n,x) = (4*n-1+(4*n-2)*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 + (4*n - 2)*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 + (4*n - 2)*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.