Real-rooted atlas

A038259 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 37\,x + 26\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112637
21215726761369
313319438223081757650653
4146411384244907767733444569761874161
51610511903330899756021266960251336801188137669343957
61771561251239561484597404678731208294114407841708163089157762565726409
71948717132242410222862799969006557560212882269603019057641623786514752803181017694931877133
821435888140533315683353210660815851541305646834099312088559024153610466066490887067992954882088270645763512479453921
923579476915015997815447423979345626155043154249273151663776219183584780163453801941990434986565126656206738793930245429503678976129961739795077
1025937424601613066399660652079716002041100782099520170007780502560482203886516352949795534047360128284072131072011370633666163205972454046873601411670956533764808584372417849
112853116706117418103435886876684951513806216493292552402938705920115680972444504474643222985051923946112388059318196492804586155578685824036133346983585280170812185740584963670344486987776177917621779460413
1231384283767218901724123063211572241359982169117523495743520484886476819087201833752494152184325056711423268144641024476600038740992151343134096632192015898672672777523201127360425887860736484485472282386432954289566616821766582952005840035281
13345227121439311060788791331698150439137679768081303805859891323367704307353903274403277832583297029472103301390503906383362441669230092332697643284136351636806912568377548051796454405373751363398802841634640711268190629888136463408026205511682481152873203736576243569224216081305397
14379749833583241125662672203908841930635600223690361825328203847852704118646333250110425765608735753641583763219885517672001978796853716723062031319347211109594996920113774081750602847999533079552206889427490853909401617782231784337856675841050768241801782439936382097542473375432704645099747032971509769012061295995008299689
Technical details
Node id
n-af2527194e932e67 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-38aa740447f30ac6

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:         11,          26;
  2:        121,         572,          676;
  3:       1331,        9438,        22308,         17576;
  4:      14641,      138424,       490776,        773344,        456976;
  5:     161051,     1903330,      8997560,      21266960,      25133680,       11881376;
  6:    1771561,    25123956,    148459740,     467873120,     829411440,      784170816,      308915776;
  7:   19487171,   322424102,   2286279996,    9006557560,   21288226960,    30190576416,    23786514752,     8031810176;
  8:  214358881,  4053331568,  33532106608,  158515413056,  468340993120,   885590241536,  1046606649088,   706799295488,   208827064576;
  9: 2357947691, 50159978154, 474239793456, 2615504315424, 9273151663776, 21918358478016, 34538019419904, 34986565126656, 20673879393024, 5429503678976;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+37*x+26*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[((11 + 37*x + 26*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] = ((11 + 37*x + 26*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.