Real-rooted atlas

A038279 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(9 + 22\,x + 13\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
191322
281234169484
372931594563219710648
4656137908821347909228561234256
5590494264651232010177957012852453712935153632
65314414605822166321353203226034701615200498224826809113379904
7478296948361131209564901504508095728733915631569393304088967627485172494357888
84304672149742877625147788127264916568131172104701515766543210947202812451789322481573072154875873536
93874204895036466357290995833969807637366821249880961430694272499829557447592418297467557266074188401106044993731207269217792
1034867844015036466357032737031320512609819471603187482144210552496904996466504257082905489240267160297333847804595440494357013785849184926559922791424
113138105960949861016934336010734452551560465159610545080104610970911619893244061316784290241421358586966122109812016977548547243044706715136479906930511792160394037584318301411328
12282429536481489544529900438891593208754187255819153260608581412248095140649926386226423702117224345562934547846823736264924458393809517007496094417407369914974247541935533225559962329808512248112855002631049216
1325418658283294773059166528941366512776583821908930840931427911558359225235205700517339856113961639593212043657223683013062852619923232641514234974692607617089528742668399565406113228693695257662725875959330277302875106592253282810057883082752
1422876792454961462619580755878434348384154129925095684417794172996856353262379612879807242757985546239582359308968671030026294235131336130183878854717988312536225371195065549053940545863102894755605135200821721717301854378074513816226343062387839373763856992896221821273427820544
Technical details
Node id
n-38deee6c8695ee46 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-421bc4a701fbf074

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:         9,         13;
  2:        81,        234,         169;
  3:       729,       3159,        4563,        2197;
  4:      6561,      37908,       82134,       79092,        28561;
  5:     59049,     426465,     1232010,     1779570,      1285245,       371293;
  6:    531441,    4605822,    16632135,    32032260,     34701615,     20049822,      4826809;
  7:   4782969,   48361131,   209564901,   504508095,    728733915,    631569393,    304088967,     62748517;
  8:  43046721,  497428776,  2514778812,  7264916568,  13117210470,  15157665432,  10947202812,   4517893224,   815730721;
  9: 387420489, 5036466357, 29099583396, 98076373668, 212498809614, 306942724998, 295574475924, 182974675572, 66074188401, 10604499373;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (9+22*x+13*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[((9 + 22*x + 13*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] = ((9 + 22*x + 13*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

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.