Real-rooted atlas

A395972 → D+1

nonnegative coefficients

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

Recurrence

\(\left(1 - x + x^{2} + 2\,n x + n^{2}\right)\,P_{n}(x) = \left(4 - 2\,x + 2\,x^{2} + 2\,x^{3} + 4\,n + 8\,n x + 4\,n x^{2} + 4\,n x^{3} + 2\,n^{2} + 6\,n^{2} x + 8\,n^{2} x^{2} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(-4\,x^{2} - 2\,x^{3} - 2\,x^{4} - 4\,n x^{2} - 4\,n x^{3} - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 20 + 32\,x + 8\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
12032860
28036028848776
327225605760307238412048
4832136006400096000384003840216672
523685990448960015360001728000552960460804414912
6640023206429352961599360037632000338688009031680645120100344960
71664081920014852096125239296511795200963379200722534400165150720103219202514108672
8419842695680663552008020131845072191488165821644802601123840016721510400334430208018579456068788307456
910342483968002695680004423680000401006592002028876595205527388160007431782400004180377600007431782400037158912002039678598144
10249856250286081016012800217451520002676326400001940871905280818313560064019108970496000224811417600001124057088000017984913408008174960640065125350672384
1159392071958528360411955297537228800156565094400015415640064000931618514534403366775789977606879229378560007193965363200003237284413440004708777328640019619905536002227019614720000
1213926402007449601216099123240606413619282418958336001058380038144008684143902720004498386541608960142246277126553602583532811059200024315602927616000994729210675200013263056142336005101175439360081181467485036544
1332276485459148803934601216015890361876483979428534681664616463335424069147495825408004863120585523200022042094053883904061956156259565568010127448619352064008665196679659520003249448754872320003999321544458240014283291230208003141846996378304512
1474055681452441600122830848000590190182400017876657111040035814856812134404846234750156800044451961602048000027355053293568000001102104702694195200027880270316804505600414304716246220800003249448754872320000011248091843788800000128549621071872000042849873690624000128634968016613572608
1516842752379165081637182504960020963131392000755443433472000183056968817049603056201114635468803544674560114688000284492554253107200001556198587367424000005642776077794279424001297699854745809715200176770012265054208000012797828942266368000004113587874299904000004387827065919897600013711959580999680005554007224762417348608
Technical details
Node id
n-a60c92d68f2a3294 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-c039e466817b8579

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     20,       32,          8;
   2:     80,      360,        288,          48;
   3:    272,     2560,       5760,        3072,          384;
   4:    832,    13600,      64000,       96000,        38400,          3840;
   5:   2368,    59904,     489600,     1536000,      1728000,        552960,         46080;
   6:   6400,   232064,    2935296,    15993600,     37632000,      33868800,       9031680,         645120;
   7:  16640,   819200,   14852096,   125239296,    511795200,     963379200,     722534400,      165150720,       10321920;
   8:  41984,  2695680,   66355200,   802013184,   5072191488,   16582164480,   26011238400,    16721510400,     3344302080,      185794560;
   9: 103424,  8396800,  269568000,  4423680000,  40100659200,  202887659520,  552738816000,   743178240000,   418037760000,    74317824000,    3715891200;
  10: 249856, 25028608, 1016012800, 21745152000, 267632640000, 1940871905280, 8183135600640, 19108970496000, 22481141760000, 11240570880000, 1798491340800, 81749606400;
%F Formula
The row polynomials satisfy (n^2+1+(2*n-1)*x+x^2)*P(n,x) = (2*n^2+4*n+4+(4*n^3+6*n^2+8*n-2)*x+(8*n^2+4*n+2)*x^2+(4*n+2)*x^3)*P(n-1,x) - ((2*n^2+4*n+4)*x^2+(4*n+2)*x^3+2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 20 + 32*x + 8*x^2.
%t Mathematica
P[1, x_] := 20 + 32*x + 8*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 4*n + 4 + (4*n^3 + 6*n^2 + 8*n - 2)*x + (8*n^2 + 4*n + 2)*x^2 + (4*n + 2)*x^3)*P[n - 1, x] - ((2*n^2 + 4*n + 4)*x^2 + (4*n + 2)*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n^2 + 1 + (2*n - 1)*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 20 + 32*x + 8*x^2; for(n = 2, N, my(i = n); v[i] = ((2*n^2 + 4*n + 4 + (4*n^3 + 6*n^2 + 8*n - 2)*x + (8*n^2 + 4*n + 2)*x^2 + (4*n + 2)*x^3)*v[i - 1] - ((2*n^2 + 4*n + 4)*x^2 + (4*n + 2)*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n^2 + 1 + (2*n - 1)*x + x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.