Real-rooted atlas

A038336 → x ↦ x/(x+1) → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(17 + 11\,x + 11\,n\right)\,P_{n}(x) = \left(336 + 748\,x + 242\,x^{2} + 132\,n + 242\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 672 + 1496\,x + 484\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
067214964842652
111232443524936810648115600
2172800988416195148814481282342564795088
32529792190080005436288071554560398235205153632192432384
43583180833393254412545280002391966720236130048010513409281133799047542280384
5495673344551809843225712805888643991040009209071872072728054784269844171522494357888290423230208
667363799048723850854448559266201615084846120962833560576000324159329894421333562736646784653455365487587353611029903530240
79029615616013338032209928636612345856320491156930567466998829875211220899880960010697257886515260343506026496167920173020161207269217792414304185935872
8119706904166419865154355200146718354309120633351572029440176270136311808032854794851450884114329956352000336199533576192016594464157286404104715340492802655992279142415422116162681856
9157270105128962896907080826882403683676979200118352805809356803831777010778112085314745974915072132514339234185216142238264205312000101700358906798080446206702895923209933411123992576584318301411328569768260120498176
102050703103098884151930775404544382391734669148162115241635741696007811285183417548802023178261690843136375384882289626316849976836511178424324693862718775296000298321052793274368011779856956452372482384018669758218241285500263104921620916275404133650432
11265699793358028858650108748627968593726100882849792364546787051254579215123977695553126400446805512491483791369643816380726352281615337153762119018086417866719052746286694414916052639663718400085319821098876469248306276280867761684485681911162923753472282810057883082752763654618781792337920
1234237825721303040818355363542728704903211674728870707260955879690639245312280701026029466025984931637026046072586240229360163078961679564842432792075195950039045904804198415821963264611438829805095144652845941442130164252672002388954990768541138944786109120893921656832134617587552347389952622182127342782054427749299709983379079168
134391001148757114881129848248803000320013502863498455023616099353284220175777792050288600744777377382401852626771794475771494451240036432533992243200108126934022939077509120175035267310183293911040216509487275246805319680201774813835681397735424137824326390492758016000656962622461348813209601995507768423031898112031731288494481884774401368800680154120519681004167640967117834747904
145608155853149437952154563240436250443776198853291789328056320015843359838187227709440874308901137546844569603540317492432327367720961086874372786092452610048257664183203599503851520047575850970093194104012806845823788131613272965120762113395208868754725273664567940427418047275335684042846907454454235136000177885263927995986345984050172766748921944866816074462757000384156270592301136149633906514329636205209121023226547339264
Technical details
Node id
n-9013281423abaed6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1b896634c6eaa14d

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 2)
  • 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:            672,            1496,              484;
  1:          11232,           44352,            49368,             10648;
  2:         172800,          988416,          1951488,           1448128,            234256;
  3:        2529792,        19008000,         54362880,          71554560,          39823520,           5153632;
  4:       35831808,       333932544,       1254528000,        2391966720,        2361300480,        1051340928,          113379904;
  5:      495673344,      5518098432,      25712805888,       64399104000,       92090718720,       72728054784,        26984417152,         2494357888;
  6:     6736379904,     87238508544,     485592662016,     1508484612096,     2833560576000,     3241593298944,      2133356273664,       678465345536,        54875873536;
  7:    90296156160,   1333803220992,    8636612345856,    32049115693056,    74669988298752,   112208998809600,    106972578865152,     60343506026496,     16792017302016,     1207269217792;
  8:  1197069041664,  19865154355200,  146718354309120,   633351572029440,  1762701363118080,  3285479485145088,   4114329956352000,   3361995335761920,   1659446415728640,   410471534049280,   26559922791424;
  9: 15727010512896, 289690708082688, 2403683676979200, 11835280580935680, 38317770107781120, 85314745974915072, 132514339234185216, 142238264205312000, 101700358906798080, 44620670289592320, 9933411123992576, 584318301411328;
%F Formula
The row polynomials satisfy (11*n+17+11*x)*P(n,x) = (132*n+336+(242*n+748)*x+242*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 672 + 1496*x + 484*x^2.
%t Mathematica
P[0, x_] := 672 + 1496*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((132*n + 336 + (242*n + 748)*x + 242*x^2)*P[n - 1, x])/(11*n + 17 + 11*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 672 + 1496*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((132*n + 336 + (242*n + 748)*x + 242*x^2)*v[i - 1])/(11*n + 17 + 11*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

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.