Real-rooted atlas

A331333 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2\right)\,P_{n}(x) = \left(4 - 2\,x + x^{2} + 4\,n x - 2\,n x^{2} + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x^{2} + 3\,x^{3} - 2\,n x^{3}\right)\,P_{n-1}'(x) + \left(x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
248214
383636686
41612828819224648
5324001600240012001205752
664115272001920021600864072058576
7128313628224117600235200211680705605040671568
82568192100352602112188160030105602257920645120403208546432
9512207363317762709504121927683048192040642560261273606531840362880119401856
10102451200103680011059200677376002438553605080320005806080003265920007257600036288001815177984
112048123904309760041817600334540800163924992049177497608781696000878169600043908480008781696003991680029808908032
124096294912892108814868480015054336009634775040393419980801011651379201580705280001405071360006322821120011496038400479001600525586164736
1381926922242492006450255462462819328005088365568027137949696094982823936021371135385602968213248000237457059840097141524480016190254080062270208009898343691264
1416384160563267837952162811084824625176576246251765760166219941888075986259148802327079186432046541583728640581769796608004231053066240015866448998400244099215360087178291200198227905206272
Technical details
Node id
n-aa53854e96f59b0e · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-9678f4e916f31c27

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:   2,     1;
  2:   4,     8,      2;
  3:   8,    36,     36,       6;
  4:  16,   128,    288,     192,       24;
  5:  32,   400,   1600,    2400,     1200,      120;
  6:  64,  1152,   7200,   19200,    21600,     8640,      720;
  7: 128,  3136,  28224,  117600,   235200,   211680,    70560,     5040;
  8: 256,  8192, 100352,  602112,  1881600,  3010560,  2257920,   645120,   40320;
  9: 512, 20736, 331776, 2709504, 12192768, 30481920, 40642560, 26127360, 6531840, 362880;
%F Formula
The row polynomials satisfy 2*P(n,x) = (4+(4*n-2)*x+(n^2-2*n+1)*x^2)*P(n-1,x) - (4*x^2+(2*n-3)*x^3)*d/dx P(n-1,x) + x^4*d^2/dx^2 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[((4 + (4*n - 2)*x + (n^2 - 2*n + 1)*x^2)*P[n - 1, x] - (4*x^2 + (2*n - 3)*x^3)*D[P[n - 1, x], x] + x^4*D[P[n - 1, x], {x, 2}])/(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] = ((4 + (4*n - 2)*x + (n^2 - 2*n + 1)*x^2)*v[i - 1] - (4*x^2 + (2*n - 3)*x^3)*deriv(v[i - 1], x) + x^4*deriv(deriv(v[i - 1], x), 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.