Real-rooted atlas

A038277 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-4 + 11\,x + 11\,n\right)\,P_{n}(x) = \left(49 + 154\,x + 121\,x^{2} + 77\,n + 121\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1181129
2203396121720
3196066996534133116524
417493862401473789583214641361584
51488629621152371600270193013176901610517663248
612269119824892317497955217520044581845173935081771561158723712
79882516944721473782583428149114051004372600686560413223216686194871713231161280
878236585869661408415677446811095578032179280509101767695776010069552724280615262421435888164895323392
96110689067745421915430482396961371735574442746155562923549754080182916698030401424122456683472613872223579476911289335387392
104721372019672175796604259982053251578435455520377227282971060415422384246507882480330458338261920019581683779354244305843802593742460125389989167104
1136156831872571286014299406666356943017181927614775477476725294809128900247898212183776847488411249339715999069323662115400263264859700155135610070998285311670611496294944500736
12274848417277477270180710437704876943734178933197054920567003611287575126053855477827220083580545376042297512205526384185614105313983510167470443592003475096148041986162732085197631384283767219640261511884800
1320761930801503930332367061134124817920793617972658009846546396861794713390162163032828246453004283555554881641027885971268196410680306737841142949201895544404514539482734336560451762499245457473439224015271434522712143931186249852409614336
1415599130675527319733734343100302635592263704717517406532674048691947333379091791970233432771724124162184509258325556609423822220739527897868049469127737027196359736392844541770919138382932035527582807118405797618740316703386997234602706123797498335832413581549956895440896
Technical details
Node id
n-2769916898bf94a3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5b73cb7ea46d8e57

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:        18,         11;
  2:       203,        396,         121;
  3:      1960,       6699,        6534,         1331;
  4:     17493,      86240,      147378,        95832,        14641;
  5:    148862,     962115,     2371600,      2701930,      1317690,       161051;
  6:   1226911,    9824892,    31749795,     52175200,     44581845,     17393508,      1771561;
  7:   9882516,   94472147,   378258342,    814911405,   1004372600,    686560413,    223216686,     19487171;
  8:  78236585,  869661408,  4156774468,  11095578032,  17928050910,  17676957760,  10069552724,   2806152624,   214358881;
  9: 611068906, 7745421915, 43048239696, 137173557444, 274615556292, 354975408018, 291669803040, 142412245668, 34726138722, 2357947691;
%F Formula
The row polynomials satisfy (11*n-4+11*x)*P(n,x) = (77*n+49+(121*n+154)*x+121*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[((77*n + 49 + (121*n + 154)*x + 121*x^2)*P[n - 1, x])/(11*n - 4 + 11*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] = ((77*n + 49 + (121*n + 154)*x + 121*x^2)*v[i - 1])/(11*n - 4 + 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. [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.