Real-rooted atlas

A200536 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 3\,x + 2\,x^{2} + 7\,n + 4\,n x\right)\,P_{n}(x) = \left(36 + 84\,x + 73\,x^{2} + 28\,x^{3} + 4\,x^{4} + 42\,n + 73\,n x + 42\,n x^{2} + 8\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
11311226
2120230157444555
3972295236392311786132810800
473443067255152555643412112968293635216199125
553136282960670680930000833345502387205250556409440880323543750
637324824183367124112126036001489284012359346736755731671249691802041602769621126461509375
725660801959552069046992148693104218445444231338520181866475107616719480430581604402839242006759207638449281281047937500
817356032152658432626310144159021273627951255363603924576352292707226603479281566223057719908112256832688701142401428156820805122014721126425617598515625
91158935041153896192540332467215806306304323560992964919580489657544830000529109098563873438846922720574107106887512344015808496119291020827537552048053952606489651379225344512292169531250
10765904896851565312044820552960148450060800346907888640607795596864828073225440897932498400786627397080561829483390328656614573157567527740617164081801960464576049884456009974887681522464001694976012761605632010244805419921875
115018692608616351887363602222046721332565344000350071305408069453274936321080340200086413500349286688137762427072601160430317508081233833070394742088234583230950359470693611624722031405044660086404685664192275402883390980736459909120458022403103232123904204878424453125000
123265173504043898443776028208680427521152821909606433633468912384745373773900801303526351001601844731088513282149332406198562086515653096161701422973914401171392017917806826968063708133692720029752140540578815124933619762720144737569944035118544665669435624192109528657921336600320120412160741580827033640961271514111328125
1321114788659230845005701122161508579942496715809755136310276926625536759728250927360147565402741785623329062308597763056138506113264336001522215902431284700740905042482339147252848168524420003613798090210316608348962015048889820926938177735276330279648480236293559650726158784005088133646220364823773172966433918739712376204441630939750417465344585728819220499921386718750
14135831217766421423891750912162599110152192790610004946944276628397604249674166220112624641584200014666214427673212335408640402663048043576324945733787320726451777769678178544465301600071890403606979280416730424196257213925706140687599578603737091695500154916309526460864020411665927354565123780222392346401046535145335682453873341920048160101749767785576547841013973309441030717542477976371240628224126156816384328906966552734375
Technical details
Node id
n-a14bebbfc041d7f8 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7dba88e059b2dbe5

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:        13,         11,          2;
  2:       120,        230,        157,          44,           4;
  3:       972,       2952,       3639,        2311,         786,         132,           8;
  4:      7344,      30672,      55152,       55564,       34121,       12968,        2936,         352,          16;
  5:     53136,     282960,     670680,      930000,      833345,      502387,      205250,       55640,        9440,         880,          32;
  6:    373248,    2418336,    7124112,    12603600,    14892840,    12359346,     7367557,     3167124,      969180,      204160,       27696,       2112,         64;
  7:   2566080,   19595520,   69046992,   148693104,   218445444,   231338520,   181866475,   107616719,    48043058,    16044028,     3924200,     675920,      76384,      4928,      128;
  8:  17356032,  152658432,  626310144,  1590212736,  2795125536,  3603924576,  3522927072,  2660347928,  1566223057,   719908112,   256832688,   70114240,   14281568,   2080512,   201472,   11264,    256;
  9: 115893504, 1153896192, 5403324672, 15806306304, 32356099296, 49195804896, 57544830000, 52910909856, 38734388469, 22720574107, 10688751234, 4015808496, 1192910208, 275375520, 48053952, 6064896, 513792, 25344, 512;
%F Formula
The row polynomials satisfy (7*n-1+(4*n+3)*x+2*x^2)*P(n,x) = (42*n+36+(73*n+84)*x+(42*n+73)*x^2+(8*n+28)*x^3+4*x^4)*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[((42*n + 36 + (73*n + 84)*x + (42*n + 73)*x^2 + (8*n + 28)*x^3 + 4*x^4)*P[n - 1, x])/(7*n - 1 + (4*n + 3)*x + 2*x^2)]];
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] = ((42*n + 36 + (73*n + 84)*x + (42*n + 73)*x^2 + (8*n + 28)*x^3 + 4*x^4)*v[i - 1])/(7*n - 1 + (4*n + 3)*x + 2*x^2)); 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.