Real-rooted atlas

A204420 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(1 - x^{2} - 4\,n - 4\,n x + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
29424596
32251365259515755760
4110258106021315024066099225645120
589302576573352529009040595310318625659823275116121600
610805602510542401104155099795852851538096604996955744963070140472832530656102400
718261468225199237503465907788686805225156355924532956849475552854696199355135766688557527392202337511158821273600
8410883035062549485112677000255058017914700738087915965400131567782080735014827063064814001033422580089900407837511081000698501159606255356234211328000
911874519713306251562834976690487589695458572019300295691147500448700618633287070148350853466058693898650777573262639473300451660534071446700151918199591783625225615874552818753278015337332736000
104286701616503556256115627602413579250386157721014940124251425329922178976814003413073844850025856505549359287294866043006212369052317154226504733011845929421054002350550231471411219256876009470847842325090020733946574681252491291656372879360000
11189043541287806830625290403946074151021612519983403628724618257175815084674931144854986752193599917991307203878504095782112759598413652505419995287835285312655505087832476324060176213503322637373662802716771251438547779465081465430253718104011123902187287543480014496195571043752301953490488540528640000
12100004033341249813400625164493691097274783193950012241043209521691432317450546084363903742183266747001629088044702809802469107753427988690494829203052310005222252222101635018244947005807960670480716887993854004683178844553957238494753752671508265193139657356071001023852675380876268329902502367919083081118328511750025001008335312453350156252541356653499348743618560000
13625025208382811333753906251095588291863311019007609375876098442085766193175756875042392976910411028170985966250138678697357428210024824656875324212736173366486857434148125557780000489753159846340112500715499566896905169231149797500684736023042070717345507254375483083838581963342183150315625244317600111841693642153698750839095498574661376654668562501754601683175457705183907812516875680626335906011355468753303763649549153366704128000000
14455643376911069462306597656258476233385847278602894780937507244603275334208076134225759375377643309692211888292149457275001342904712944170001659791864356253449365047012504573685072003162506604801927807760127373528799943759583409246513205925291925104650001059408721677946300452865079881875888316424368192648133048760266250556362176091209891594536743523125252470742733664860479931684327500784922238087450724628512363968751496876829692372137981781184375013213657930421014406891332031254995290638118319890456641536000000
Technical details
Node id
n-ab955e73adb3d5d4 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-438412a981173c5d

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:                9,                42,                45;
  3:              225,              1365,              2595,               1575;
  4:            11025,             81060,            213150,             240660,              99225;
  5:           893025,           7657335,          25290090,           40595310,           31862565,            9823275;
  6:        108056025,        1054240110,        4155099795,         8528515380,         9660499695,         5744963070,         1404728325;
  7:      18261468225,      199237503465,      907788686805,      2251563559245,      3295684947555,      2854696199355,      1357666885575,       273922023375;
  8:    4108830350625,    49485112677000,   255058017914700,    738087915965400,   1315677820807350,   1482706306481400,   1033422580089900,    407837511081000,     69850115960625;
  9: 1187451971330625, 15628349766904875, 89695458572019300, 295691147500448700, 618633287070148350, 853466058693898650, 777573262639473300, 451660534071446700, 151918199591783625, 22561587455281875;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4*n^2-4*n+1+(8*n^2-4*n)*x+(4*n^2-1)*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[((4*n^2 - 4*n + 1 + (8*n^2 - 4*n)*x + (4*n^2 - 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] = ((4*n^2 - 4*n + 1 + (8*n^2 - 4*n)*x + (4*n^2 - 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.