Real-rooted atlas

A038277 → x ↦ x/(x+1) → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(25 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1251843
26259003241849
3156253375024300583279507
4390625112500012150005832001049763418801
59765625351562505062500036450000131220001889568147008443
6244140625105468750018984375001822500000984150000283435200340122246321363049
7610351562530761718750664453125007973437500057408750000248005800005952139200612220032271818611107
815258789062587890625000022148437500003189375000000287043750000016533720000005952139200001224440064001101996057611688200277601
9381469726562524719238281250711914062500001196015625000001291696875000009300217500000044641044000000137749507200002479491129600198359290368502592611936843
109536743164062568664550781250022247314453125004271484375000000538207031250000046501087500000002790065250000000114791256000000030993639120000049589822592000357046722662421611482313284249
11238418579101562518882751464843750679779052734375001468322753906250002114384765625000002131299843750000001534535887500000007891898850000000028410835860000000681860060640000098187848732160064268410079232929293739471222707
1259604644775390625514984130859375000203933715820312500048944091796875000007928942871093750000913414218750000000076726794375000000004735139310000000000213081268950000000068186006064000000014728177309824000019280523023769600115683138142617639959630797262576401
131490116119384765625139474868774414062506025314331054687500015906829833984375000028632293701171875000037107452636718750000035623154531250000000025648671262500000000013850282481750000000055401129927000000000159555254189760000003133084991362560000375970198963507200208229648656711681718264124282290785243
1437252902984619140625375509262084960937500175738334655761718750050612640380859375000001002130279541015625000014430676025390625000000155851301074218750000001282433563125000000000080793314476875000000003878079094890000000000139610847416040000000036552658232563200000065794784818613760000728803770298490880037481336758208102473885357344138503765449
Technical details
Node id
n-59229dc0e932fb45 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-9fb853a77717bbd4

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:            25,             18;
  2:           625,            900,            324;
  3:         15625,          33750,          24300,            5832;
  4:        390625,        1125000,        1215000,          583200,          104976;
  5:       9765625,       35156250,       50625000,        36450000,        13122000,        1889568;
  6:     244140625,     1054687500,     1898437500,      1822500000,       984150000,      283435200,       34012224;
  7:    6103515625,    30761718750,    66445312500,     79734375000,     57408750000,    24800580000,     5952139200,      612220032;
  8:  152587890625,   878906250000,  2214843750000,   3189375000000,   2870437500000,  1653372000000,   595213920000,   122444006400,   11019960576;
  9: 3814697265625, 24719238281250, 71191406250000, 119601562500000, 129169687500000, 93002175000000, 44641044000000, 13774950720000, 2479491129600, 198359290368;
%F Formula
The row polynomials satisfy P(n,x) = (25+18*x)*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[(25 + 18*x)*P[n - 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] = (25 + 18*x)*v[i - 1]); 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.