Real-rooted atlas

A038326 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(23 + 58\,x + 35\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1233558
2529161012253364
312167555458452542875195112
4279841170338038881503944500150062511316496
564363434897217514904575022680875017257187552521875656356768
61480358891351632030514207837510433202500119074593757248018750183826562538068692544
7340482544736268792805165574923675419936400625639033653125583465509375295960765625643392968752207984167552
878310985281953351125160507763099270015453659543000293955480437503578588457500027228390437500118384306250002251875390625128063081718016
91801152661463246679603635151501528022127005331512542335001216975689011250185191952675625018787589401875001225277569687500466138205859375788156386718757427658739644928
104142651121364963040343151205043168930636151251751782692481500046650734745431250851882982307875001080286390607812509393794700937500053605893673828125181275968945312502758547353515625430804206899405824
11952809757913927159492068172548651213526605660696255540012764972743751686090841513443750359210657539820625054662491364755312505941575148342968750452076369982617187522931410071582031256979124804394531259654915737304687524986644000165537792
122191462443202032140018009832384934033493334316235216501698937247924974750058170134032213809375141631630687129275000251447460277874437500327974948188531875000311932695288005859375210968972658554687500963119223006445312502664756743496093750033792205080566406251449225352009601191936
13504036361936467383997115411656924605591040972368675724850507982237129567450250193254111951465877812552934821969314566531251074039866044063668750016344084918061838437500186535751782227503906251576993070622696289062595990882559642382812503983811331526660156250101038693190893554687511827271778198242187584055070416556869132288
141159283632453874980924697781734886901767024429327585594652834751487002548688370172550062227824048372012655625189389029712436560256250432301046082735626671875751827906230844568125000100107520123128760429687510155835374810164101562507727266046051211816406254275957495838615234375001626722960373386230468753808381512579833984375041395451223693847656254875194084160298409672704
Technical details
Node id
n-6a1852cf3e67c87f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-86ce3654926e25ab

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:            23,             35;
  2:           529,           1610,            1225;
  3:         12167,          55545,           84525,           42875;
  4:        279841,        1703380,         3888150,         3944500,          1500625;
  5:       6436343,       48972175,       149045750,       226808750,        172571875,         52521875;
  6:     148035889,     1351632030,      5142078375,     10433202500,      11907459375,       7248018750,       1838265625;
  7:    3404825447,    36268792805,    165574923675,    419936400625,     639033653125,     583465509375,     295960765625,      64339296875;
  8:   78310985281,   953351125160,   5077630992700,  15453659543000,   29395548043750,   35785884575000,   27228390437500,   11838430625000,   2251875390625;
  9: 1801152661463, 24667960363515, 150152802212700, 533151254233500, 1216975689011250, 1851919526756250, 1878758940187500, 1225277569687500, 466138205859375, 78815638671875;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (23+58*x+35*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[((23 + 58*x + 35*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] = ((23 + 58*x + 35*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

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.