Real-rooted atlas

A038327 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(169 - 26\,x + 78\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2197 + 1014\,x + 156\,x^{2} + 8\,x^{3} + 1014\,n x + 260\,n x^{2} + 16\,n x^{3} + 52\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
113821
216920836413
32197405614041287785
42856170304365046656400142425
537129311424407909202163202600011522548125
6482680917822064154229405624320101400089856313644803125
7627485172703013042806975081279532803075800040884482853768192776840625

All 15 rows

Technical details
Node id
n-4e8f8f60ddebe9dc · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8403dd4a9de0b7a8

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:          13,           8;
  2:         169,         208,          36;
  3:        2197,        4056,        1404,         128;
  4:       28561,       70304,       36504,        6656,         400;
  5:      371293,     1142440,      790920,      216320,       26000,       1152;
  6:     4826809,    17822064,    15422940,     5624320,     1014000,      89856,      3136;
  7:    62748517,   270301304,   280697508,   127953280,    30758000,    4088448,    285376,     8192;
  8:   815730721,  4015905088,  4865423472,  2661428224,   799708000,  141732864,  14839552,   851968,   20736;
  9: 10604499373, 58732611912, 81322078032, 51897850368, 18713167200, 4145686272, 578742528, 49840128, 2426112, 51200;
%F Formula
The row polynomials satisfy (169+(78*n-26)*x+4*n^2*x^2)*P(n,x) = (2197+(1014*n+1014)*x+(52*n^2+260*n+156)*x^2+(8*n^2+16*n+8)*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[((2197 + (1014*n + 1014)*x + (52*n^2 + 260*n + 156)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(169 + (78*n - 26)*x + 4*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] = ((2197 + (1014*n + 1014)*x + (52*n^2 + 260*n + 156)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(169 + (78*n - 26)*x + 4*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

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.