Real-rooted atlas

A130123 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,x + 12\,n x + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 24\,x + 96\,x^{2} + 128\,x^{3} + 24\,n x + 160\,n x^{2} + 256\,n x^{3} + 32\,n^{2} x^{2} + 128\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
123234
24128576708
383843456819212040
41610241382465536102400182800
532256046080327680102400011796482580000
664614413824013107206144000141557761284505634600000
71281433638707245875202867200099090432179830784134217728446800000
8256327681032192146800641146880005284823041438646272214748364813589544965604000000
9512737282654208440401924128768002378170368863187763219327352832244611809281342177280068680000000
10102416384066355201258291201376256000951268147243159388160128849018880244611809280268435456000129922760704826000000000
1120483604481622016034603008043253760003487983206418990130790470866960384017938199347202952790016000285830073548812369505812489780000000000
124096786432389283849227468801297612800011958799564875960523161634016140984321076291960832023622320128000342996088258562968681394995211613591568384114280000000000
138192170393692012544239914188837486592000388660985856282139086028814740327759872559671819632641535450808320002972632764907523859285813493763019533807779841077521395220481320400000000000
1416384367001621469593661069066241049624576001209167511552987486801100858961311039488261180182495232859852452659200208084293543526436020000925941764227347330891776301705990661734498956046499840015108000000000000
Technical details
Node id
n-cb27801678970979 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6fb0656952daab5a

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   2,    32;
  2:   4,   128,     576;
  3:   8,   384,    3456,     8192;
  4:  16,  1024,   13824,    65536,    102400;
  5:  32,  2560,   46080,   327680,   1024000,    1179648;
  6:  64,  6144,  138240,  1310720,   6144000,   14155776,   12845056;
  7: 128, 14336,  387072,  4587520,  28672000,   99090432,  179830784,   134217728;
  8: 256, 32768, 1032192, 14680064, 114688000,  528482304, 1438646272,  2147483648,  1358954496;
  9: 512, 73728, 2654208, 44040192, 412876800, 2378170368, 8631877632, 19327352832, 24461180928, 13421772800;
%F Formula
The row polynomials satisfy (1+(12*n-4)*x+16*n^2*x^2)*P(n,x) = (2+(24*n+24)*x+(32*n^2+160*n+96)*x^2+(128*n^2+256*n+128)*x^3)*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[((2 + (24*n + 24)*x + (32*n^2 + 160*n + 96)*x^2 + (128*n^2 + 256*n + 128)*x^3)*P[n - 1, x])/(1 + (12*n - 4)*x + 16*n^2*x^2)]];
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] = ((2 + (24*n + 24)*x + (32*n^2 + 160*n + 96)*x^2 + (128*n^2 + 256*n + 128)*x^3)*v[i - 1])/(1 + (12*n - 4)*x + 16*n^2*x^2)); 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.