Real-rooted atlas

A038327 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(169 - 13\,x + 39\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(2197 + 507\,x + 39\,x^{2} + x^{3} + 507\,n x + 65\,n x^{2} + 2\,n x^{3} + 13\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
113417
21691049282
321972028351164592
4285613515291268322573696
5371293571220197730270401625361168944
64826809891103238557357030406337528084918362848
76274851713515065270174377159941601922375127764445964286122368
8815730721200795254412163558683326785284998175044291522318686656814427367168
9106044993732936630595620330519508648723129611695729501295526969042852389376947710068097123584
1013785849184942417997492033037094200512047715264025340747250336837009629389269016872960616005130001211041907073536
111792160394037606577364135652492271896352153529103440517675265250802794872888405330934603208320293629059295001730314415867683930112
122329808512248186023698913776818879441583063732783779296010094667672375178908571670421853860428418820099584114515329548334000134963422464169240659872940032
1330287510659225312115004263690121258096596613974630840458701024189555426292375377944357653725276146303428530099471616387061813712042111500760293821898208285611963636509522358272
143937376385699289169610059691661681908113171531193910437542134871488344990875852122576428747881085612003232840298713782586262016117408750158774332858600345933688111515795225990513567222554766319354535936
Technical details
Node id
n-8e34fab1b183390d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-74680266075103d3

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,           4;
  2:         169,         104,           9;
  3:        2197,        2028,         351,         16;
  4:       28561,       35152,        9126,        832,         25;
  5:      371293,      571220,      197730,      27040,       1625,        36;
  6:     4826809,     8911032,     3855735,     703040,      63375,      2808,      49;
  7:    62748517,   135150652,    70174377,   15994160,    1922375,    127764,    4459,     64;
  8:   815730721,  2007952544,  1216355868,  332678528,   49981750,   4429152,  231868,   6656,   81;
  9: 10604499373, 29366305956, 20330519508, 6487231296, 1169572950, 129552696, 9042852, 389376, 9477, 100;
%F Formula
The row polynomials satisfy (169+(39*n-13)*x+n^2*x^2)*P(n,x) = (2197+(507*n+507)*x+(13*n^2+65*n+39)*x^2+(n^2+2*n+1)*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 + (507*n + 507)*x + (13*n^2 + 65*n + 39)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(169 + (39*n - 13)*x + 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 + (507*n + 507)*x + (13*n^2 + 65*n + 39)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(169 + (39*n - 13)*x + 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.