Real-rooted atlas

A038335 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16 - 12\,x + 36\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(192 + 432\,x + 324\,x^{2} + 81\,x^{3} + 432\,n x + 540\,n x^{2} + 162\,n x^{3} + 108\,n^{2} x^{2} + 81\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123648
21448647291737
3172815552262441166455188
4207362488326298565598721640251623321
5248832373248012597120167961609841500212576445341856
6298598453747712226748160403107840354294000153055008260406091219979313
7358318087524679683809369088846526464099202320006428310336218741115630611001631904997012

All 15 rows

Technical details
Node id
n-6fdf8b41f34cb1ba · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e997b32cda4979c2

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:         12,           36;
  2:        144,          864,          729;
  3:       1728,        15552,        26244,         11664;
  4:      20736,       248832,       629856,        559872,        164025;
  5:     248832,      3732480,     12597120,      16796160,       9841500,       2125764;
  6:    2985984,     53747712,    226748160,     403107840,     354294000,     153055008,      26040609;
  7:   35831808,    752467968,   3809369088,    8465264640,    9920232000,    6428310336,    2187411156,     306110016;
  8:  429981696,  10319560704,  60949905408,  162533081088,  238085568000,  205705930752,  104995735488,   29386561536,   3486784401;
  9: 5159780352, 139314069504, 940369969152, 2925595459584, 5142648268800, 5554060130304, 3779846477568, 1586874322944, 376572715308, 38742048900;
%F Formula
The row polynomials satisfy (16+(36*n-12)*x+9*n^2*x^2)*P(n,x) = (192+(432*n+432)*x+(108*n^2+540*n+324)*x^2+(81*n^2+162*n+81)*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[((192 + (432*n + 432)*x + (108*n^2 + 540*n + 324)*x^2 + (81*n^2 + 162*n + 81)*x^3)*P[n - 1, x])/(16 + (36*n - 12)*x + 9*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] = ((192 + (432*n + 432)*x + (108*n^2 + 540*n + 324)*x^2 + (81*n^2 + 162*n + 81)*x^3)*v[i - 1])/(16 + (36*n - 12)*x + 9*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.