Real-rooted atlas

A038208 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,n x\right)\,P_{n}(x) = \left(4 + 16\,x + 16\,x^{2} + 8\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141620
216128192336
364768230420485184
4256409618432327682048076032
51024204801228803276804096001966081078272
6409698304737280262144049152004718592183500814929920
71638445875241287681835008045875200660602885138022416777216203046912
8655362097152220200961174405123670016007046430728220835845368709121509949442723217408
9262144943718411324620870464307226424115206341787648986500300896636764165435817984134217728036118462464
1010485764194304056623104040265318401761607680050734301184986500300801288490188801087163596805368709120011811160064474699792384
1141943041845493762768240640221459251201107296256003720515420168681202647041417339207680159450660864011811160064005196910428161030792151046191736422400
1216777216805306368132875550721181116006406643777536002551210573824694496211763213606456393728191340793036801889785610240012472585027584494780232499289335319756880244904034304
1367108864348966092862813896704614180323328383862702080016582868729856515911471595521179226220789761989944247582722456721293312002161914738114561286428604497924645436627353676965813944321034267651997696
1426843545615032385536293131517952312673619148821496311316480103182294319104361138030116864943380976631808185728129774387227515278485094403026680633360384240133339506278413007222556590084310085580881926597069766656013267157467004928
Technical details
Node id
n-996c83edfd020e5d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7cabb77b45806852

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:      4,      16;
  2:     16,     128,       192;
  3:     64,     768,      2304,      2048;
  4:    256,    4096,     18432,     32768,      20480;
  5:   1024,   20480,    122880,    327680,     409600,     196608;
  6:   4096,   98304,    737280,   2621440,    4915200,    4718592,    1835008;
  7:  16384,  458752,   4128768,  18350080,   45875200,   66060288,   51380224,   16777216;
  8:  65536, 2097152,  22020096, 117440512,  367001600,  704643072,  822083584,  536870912,  150994944;
  9: 262144, 9437184, 113246208, 704643072, 2642411520, 6341787648, 9865003008, 9663676416, 5435817984, 1342177280;
%F Formula
The row polynomials satisfy (1+2*n*x)*P(n,x) = (4+(8*n+16)*x+(16*n+16)*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[((4 + (8*n + 16)*x + (16*n + 16)*x^2)*P[n - 1, x])/(1 + 2*n*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] = ((4 + (8*n + 16)*x + (16*n + 16)*x^2)*v[i - 1])/(1 + 2*n*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.