Real-rooted atlas

A013611 → D → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
286472
312192576780
416384230440966800
5206405760204802560052500
6249601152061440153600147456375000
72813442016014336053760010321928028162537500
832179232256286720143360041287686422528419430416500000
936230448384516096322560012386304289013763774873621233664104062500
104028806912086016064512003096576096337920188743680212336640104857600640625000
114435209504013516801182720068124672264929280692060160116785152011534336005075107843867187500
124842241267202027520202752001362493446358302722076180480467140608069206016006090129408241591910422968750000
13524992164736292864032947200253034496137763225653980692481518206976029989273600395858411523140694835211341398016134570312500
145658242096644100096512512004428103682755264512125954949124250979532810496245760018473392537621984863846415877957222452613349376779296875000
Technical details
Node id
n-9e35a72c363d5430 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-14ef8ed0e797c5a3

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:  8,   64;
  3: 12,  192,   576;
  4: 16,  384,  2304,   4096;
  5: 20,  640,  5760,  20480,   25600;
  6: 24,  960, 11520,  61440,  153600,   147456;
  7: 28, 1344, 20160, 143360,  537600,  1032192,   802816;
  8: 32, 1792, 32256, 286720, 1433600,  4128768,  6422528,  4194304;
  9: 36, 2304, 48384, 516096, 3225600, 12386304, 28901376, 37748736, 21233664;
%F Formula
The row polynomials satisfy (n-1+(4*n^2-8*n+4)*x)*P(n,x) = (n+(4*n^2+4*n)*x+16*n^2*x^2)*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[((n + (4*n^2 + 4*n)*x + 16*n^2*x^2)*P[n - 1, x])/(n - 1 + (4*n^2 - 8*n + 4)*x)]];
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] = ((n + (4*n^2 + 4*n)*x + 16*n^2*x^2)*v[i - 1])/(n - 1 + (4*n^2 - 8*n + 4)*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.