Real-rooted atlas

A038291 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(10 + 24\,x + 14\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1101424
2100280196576
3100042005880274413824
4100005600011760010976038416331776
51000007000001960000274400019208005378247962624
610000008400000294000005488000057624000322694407529536191102976
71000000098000000411600000960400000134456000011294304005270675201054135044586471424
8100000000112000000054880000001536640000026891200000301181440002108270080084330803201475789056110075314176
910000000001260000000070560000000230496000000484041600000677658240000632481024000379488614400132821015040206610467842641807540224
1010000000000140000000000882000000000329280000000080673600000001355316480000015812025600000126496204800006641050752000206610467840028925465497663403380965376
11100000000000154000000000010780000000000452760000000001267728000000002484746880000003478645632000003478645632000002435051942400001136357573120003181801204736040495651696641521681143169024
1210000000000001680000000000012936000000000060368000000000019015920000000004259566080000000695729126400000083487495168000007305155827200000454543029248000019090807228416004859478203596805669391237529636520347436056576
131000000000000018200000000000015288000000000007847840000000000274674400000000006921794880000000012920683776000000018088957286400000018993405150720000014772648450560000082726831323136000315866083233792007370208608788480793714773254144876488338465357824
141000000000000001960000000000000178360000000000009988160000000000038454416000000000010767236480000000002261119660800000000361779145728000000044317945351680000004136341566156800000289543909630976000014740417217576960005159146026151936001111200682555801601111200682555801621035720123168587776
Technical details
Node id
n-6a2532ecdfb2c281 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f96e0317bce1f626

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:         10,          14;
  2:        100,         280,         196;
  3:       1000,        4200,        5880,         2744;
  4:      10000,       56000,      117600,       109760,        38416;
  5:     100000,      700000,     1960000,      2744000,      1920800,       537824;
  6:    1000000,     8400000,    29400000,     54880000,     57624000,     32269440,      7529536;
  7:   10000000,    98000000,   411600000,    960400000,   1344560000,   1129430400,    527067520,    105413504;
  8:  100000000,  1120000000,  5488000000,  15366400000,  26891200000,  30118144000,  21082700800,   8433080320,   1475789056;
  9: 1000000000, 12600000000, 70560000000, 230496000000, 484041600000, 677658240000, 632481024000, 379488614400, 132821015040, 20661046784;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (10+24*x+14*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[((10 + 24*x + 14*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] = ((10 + 24*x + 14*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.