Real-rooted atlas

A038259 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 - 55\,x + 165\,n x + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 1815\,x + 825\,x^{2} + 125\,x^{3} + 1815\,n x + 1375\,n x^{2} + 250\,n x^{3} + 275\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112031
2121440225786
3133172607425200018016
4146411064801633508800015625388096
51610511464100299475024200008593751125008011776
61771561193261204941337553240000283593757425000765625160301056
71948717124801854076096597510248700007278906252858625005895312550000003131047936

All 15 rows

Technical details
Node id
n-2884f9ccb99483cb · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-44310a99e9adba32

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:         11,          20;
  2:        121,         440,          225;
  3:       1331,        7260,         7425,         2000;
  4:      14641,      106480,       163350,        88000,        15625;
  5:     161051,     1464100,      2994750,      2420000,       859375,       112500;
  6:    1771561,    19326120,     49413375,     53240000,     28359375,      7425000,      765625;
  7:   19487171,   248018540,    760965975,   1024870000,    727890625,    285862500,    58953125,     5000000;
  8:  214358881,  3117947360,  11160834300,  18037712000,  16013593750,   8385300000,  2593937500,   440000000,   31640625;
  9: 2357947691, 38584598580, 157846085100, 297622248000, 317069156250, 207536175000, 85599937500, 21780000000, 3132421875, 195312500;
%F Formula
The row polynomials satisfy (121+(165*n-55)*x+25*n^2*x^2)*P(n,x) = (1331+(1815*n+1815)*x+(275*n^2+1375*n+825)*x^2+(125*n^2+250*n+125)*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[((1331 + (1815*n + 1815)*x + (275*n^2 + 1375*n + 825)*x^2 + (125*n^2 + 250*n + 125)*x^3)*P[n - 1, x])/(121 + (165*n - 55)*x + 25*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] = ((1331 + (1815*n + 1815)*x + (275*n^2 + 1375*n + 825)*x^2 + (125*n^2 + 250*n + 125)*x^3)*v[i - 1])/(121 + (165*n - 55)*x + 25*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.