Real-rooted atlas

A038324 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 10\,x + 10\,n\right)\,P_{n}(x) = \left(121 + 220\,x + 100\,x^{2} + 110\,n + 100\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1211031
2341420100861
34961102306300100022491
4678811984402046008400010000564921
5893101339405049610003410000105000010000013808151
611434621535860601018215009922000051150000126000001000000330812181
7143496441800423470187551210023758350001736350000716100000147000000100000007804717011
81773332561114797152803201693880050013656000475167000002778160000095480000001680000000100000000181909942641
92165024698115959993049051658718760096050816400011253072600008553006000004167240000001227600000001890000000010000000004198337389071
102617321937012165024698100797999652450017219572920000240127041000002250614520000014255010000000595320000000015345000000002100000000001000000000096107885636301
11313842837672128790541307110119076358395500292599872565000473538255300000528279490200000412612662000000224007300000000818565000000001875500000000023100000000001000000000002185064408144331
1237375828850041376611405206520172743247842660047630543358200008777996176950000113649181272000001056558980400000070733599200000003360109500000000109142000000000022506000000000025200000000000100000000000049389127576453161
134425184011176614858857750505330244797413384238007485540739848600015479926591415000022822790060070000024623989275600000019621809636000000011494209870000000048534915000000000141884600000000002659800000000000273000000000000100000000000001110729954219382791
14521292953373358161952576156472540340120042535373100114238792912644400026199392589470100004334379445596200000532531768068300000049247978551200000003433816686300000000178798820200000000067948881000000000018058040000000000031031000000000000294000000000000010000000000000024870052815998233221
Technical details
Node id
n-c0148382aea5164e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3e3a8419375cf2f4

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,           10;
  2:         341,          420,          100;
  3:        4961,        10230,         6300,         1000;
  4:       67881,       198440,       204600,        84000,         10000;
  5:      893101,      3394050,      4961000,      3410000,       1050000,       100000;
  6:    11434621,     53586060,    101821500,     99220000,      51150000,     12600000,      1000000;
  7:   143496441,    800423470,   1875512100,   2375835000,    1736350000,    716100000,    147000000,     10000000;
  8:  1773332561,  11479715280,  32016938800,  50013656000,   47516700000,  27781600000,   9548000000,   1680000000,   100000000;
  9: 21650246981, 159599930490, 516587187600, 960508164000, 1125307260000, 855300600000, 416724000000, 122760000000, 18900000000, 1000000000;
%F Formula
The row polynomials satisfy (10*n+1+10*x)*P(n,x) = (110*n+121+(100*n+220)*x+100*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[((110*n + 121 + (100*n + 220)*x + 100*x^2)*P[n - 1, x])/(10*n + 1 + 10*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] = ((110*n + 121 + (100*n + 220)*x + 100*x^2)*v[i - 1])/(10*n + 1 + 10*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

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.