Real-rooted atlas

A038335 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(21 + 69\,x + 48\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1214869
2441201623044761
3926163504145152110592328509
4194481177811260963849289728530841622667121
54084101466754402133734404877107205573836802548039681564031349
6857661211176221088672126336020483850240351151718403210529996812230590464107918163081
71801088541288174166561976051427847527814963201720643420160235973954764817978967982085870683422727446353252589
837822859361691617999744553294399795225293458276352722670236467201321454146682881510233310494729862748150169628179280429056513798374428641
97942800465811633947524395214938948794470479674393570508827316934938460166243870843076608951446985611673693202970019102725325884001091584135260546059468835452087835576229
101667988097820138125442235888039214740585484802390231807115264095609272284610560262242575409217536499509667446128640652420790133719040559217820114616320284047146724884480649250621085450242446194060654759801
11350277500542221880697715649012810065116750274432069017943430453248031551059853921484801009633915325487513623077346636011143168376773006302222745604305977214882545664032807445446724157440149976893470739005443116402981210161152168787390185178426269
12735582751138664120175984031231929625364094210691568641932502416052690944099385838539852677120363468209517175504896969248558712468013056189893595176320263782427127656453760037683202755825417524829224960188970885773131146854478533355126496061030414958734309808773529611646329922777311412561
131544723777391194614590036367105263984629490701774436203525275731595823846277123014703769042197872640124033526497736141045763780069378978625250918486401585805225720020992148117004237529805750272188085084746069594603520171963506053549343637504107198029747667123306496408373446657779517358087180192468708211294208803596764671634487466709
143243919932521508681103805437840688277792154225221934736869862414100591719747370958848886322908098406174556164051761865592713940828161389175496774644779712512362886660381948024088166472577332076389604817633281105940298306889216268697612639317694935876757356544105054069152713780840366086003089665869358905163776211097658580021412049715234464923849799414212198455448176762342779635202921
Technical details
Node id
n-e056d449033a10a5 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5dbcc0e32705bbae

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:           21,             48;
  2:          441,           2016,            2304;
  3:         9261,          63504,          145152,          110592;
  4:       194481,        1778112,         6096384,         9289728,          5308416;
  5:      4084101,       46675440,       213373440,       487710720,        557383680,        254803968;
  6:     85766121,     1176221088,      6721263360,     20483850240,      35115171840,      32105299968,      12230590464;
  7:   1801088541,    28817416656,    197605142784,    752781496320,    1720643420160,    2359739547648,    1797896798208,     587068342272;
  8:  37822859361,   691617999744,   5532943997952,  25293458276352,   72267023646720,  132145414668288,  151023331049472,   98627481501696,   28179280429056;
  9: 794280046581, 16339475243952, 149389487944704, 796743935705088, 2731693493846016, 6243870843076608, 9514469856116736, 9320297001910272, 5325884001091584, 1352605460594688;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (21+69*x+48*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[((21 + 69*x + 48*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] = ((21 + 69*x + 48*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

Candidate equalities ?

Same fingerprint ?

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.