Real-rooted atlas

A038335 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 - 63\,x + 189\,n x + 81\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1029 + 3969\,x + 5103\,x^{2} + 2187\,x^{3} + 3969\,n x + 8505\,n x^{2} + 4374\,n x^{3} + 1701\,n^{2} x^{2} + 2187\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
121108129
24414536656111538
39261142884413343314928880416
4194481400075217360406264539521328602561295616
54084101105019740607614210138883248013950326255165606524017143808
68576612126464974481913984761558330964160878870553756508664215218983603961252160376832
7180108854164839187476562711519881214366293288043064657133754783868198172279058978226766946260499215323401027584

All 15 rows

Technical details
Node id
n-6a7de36842666def · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-238fadf8fd88f70e

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:           21,            108;
  2:          441,           4536,            6561;
  3:         9261,         142884,          413343,           314928;
  4:       194481,        4000752,        17360406,         26453952,         13286025;
  5:      4084101,      105019740,       607614210,       1388832480,       1395032625,         516560652;
  6:     85766121,     2646497448,     19139847615,      58330964160,      87887055375,       65086642152,       18983603961;
  7:   1801088541,    64839187476,    562711519881,    2143662932880,    4306465713375,     4783868198172,     2790589782267,      669462604992;
  8:  37822859361,  1556140499424,  15755922556668,   72027074544768,  180871559961750,   267896619097632,   234409541710428,   112469717638656,   22876792454961;
  9: 794280046581, 36763819298892, 425409909030036, 2268852848160192, 6836944966554150, 12658115252363112, 14767801127756964, 10628388316852992, 4323713773987629, 762559748498700;
%F Formula
The row polynomials satisfy (49+(189*n-63)*x+81*n^2*x^2)*P(n,x) = (1029+(3969*n+3969)*x+(1701*n^2+8505*n+5103)*x^2+(2187*n^2+4374*n+2187)*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[((1029 + (3969*n + 3969)*x + (1701*n^2 + 8505*n + 5103)*x^2 + (2187*n^2 + 4374*n + 2187)*x^3)*P[n - 1, x])/(49 + (189*n - 63)*x + 81*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] = ((1029 + (3969*n + 3969)*x + (1701*n^2 + 8505*n + 5103)*x^2 + (2187*n^2 + 4374*n + 2187)*x^3)*v[i - 1])/(49 + (189*n - 63)*x + 81*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.