Real-rooted atlas

A013622 → x ↦ x/(x+1) → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(3\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 8\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
188
248128176
3216115215362904
48646912184321638442592
5324034560138240245760163840585640
6116641555208294402211840294912015728647730448
74082465318443545601548288030965760330301441468006499207416
8139968261273620901888928972802477260803963617283523215361342177281247178944
947239210077696940584965016453121672151040356725555247563407363623878656120795955215433839432
101574640377913604031078402508226560100329062402675441664047563407360543581798403623878656010737418240188635815280
11519631213856832016628198401182449664055180984320176579149824392398110720597939978240597939978240354334801920944892805122282493364888
121700611249884595266512793605321023488028378791936010594748989442825266397184538145980416071752797388806378026434560340161409843282463372083227389920378656
1355269864176863564825939989504230577684480138346610688059027887226881836423158169641975386472448699589774540808291434364928066331474919424321607151124487146825580544326396551178984
14178564176619022476899043596288968426274816645617516544030989640794112110185389490176293827705307136587655410614272870600608317440928640648871936675375017361408300166674382848615726511554563866543760120272
Technical details
Node id
n-efaf6bbff08d2b38 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-10cba46fcb583bff

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:      0;
  1:      8;
  2:     48,      128;
  3:    216,     1152,     1536;
  4:    864,     6912,    18432,     16384;
  5:   3240,    34560,   138240,    245760,     163840;
  6:  11664,   155520,   829440,   2211840,    2949120,    1572864;
  7:  40824,   653184,  4354560,  15482880,   30965760,   33030144,   14680064;
  8: 139968,  2612736, 20901888,  92897280,  247726080,  396361728,  352321536,  134217728;
  9: 472392, 10077696, 94058496, 501645312, 1672151040, 3567255552, 4756340736, 3623878656, 1207959552;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (3*n+8*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 8.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 8*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 8; for(n = 2, N, my(i = n + 1); v[i] = ((3*n + 8*n*x)*v[i - 1])/(n - 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.