Real-rooted atlas

A130123 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(2\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2163248
348192192432
4128768153610243456
53202560768010240512025920
6768768030720614406144024576186624
71792215041075202867204300803440641146881306368
840965734434406411468802293760275251218350085242888957952
99216147456103219241287681032192016515072165150729437184235929660466176
10204803686402949120137625604128768082575360110100480943718404718592010485760403107840
1145056901120811008043253760151388160363331584605552640692060160519045120230686720461373442660511744
129830421626882162688012976128051904512014533263362906652672415236096041523609602768240640110729625620132659217414258688
132129925111808562298883748659201686896640539806924812595494912215922769922699034624023991418880143948513285234491392872415232113192681472
144587521192755214313062410496245765248122880188932423685038197964810076395929615114593894416793993216013435194572873282879488244276264963758096384731398864896
Technical details
Node id
n-a487462ee8626f77 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-11aec07ce60b6d61

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:    4;
  2:   16,     32;
  3:   48,    192,     192;
  4:  128,    768,    1536,    1024;
  5:  320,   2560,    7680,   10240,     5120;
  6:  768,   7680,   30720,   61440,    61440,    24576;
  7: 1792,  21504,  107520,  286720,   430080,   344064,   114688;
  8: 4096,  57344,  344064, 1146880,  2293760,  2752512,  1835008,  524288;
  9: 9216, 147456, 1032192, 4128768, 10321920, 16515072, 16515072, 9437184, 2359296;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n+4*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 4*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 4*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.