Real-rooted atlas

A038207 → basis A008277 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 8\,x + 6\,x^{2} + x^{3} - n x - 2\,n x^{2} - n x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
29251751
32711816777389
481499110810613723121
52431996621493016754191526423
672977413185166464751414399610481236403
7218729506154881423781664439603422296015608142235045
865611114637295902520531514801164544474894772206543137293922303745
91968341960833725361434294536699599596249656218758640449145149670952409837234492999
105904915800171542670979328270247632386501040847674315104601304692342117592112657799163607862591823251
1117714759657747022298743089604916100130303948293658658720500875537333095876470632296915543288034862211639320530048874853
12531441226230673193374482315049593102071882782974919390259859634836847835961828458128507558303167636264758299427135726092865549453364618712945
131594323862407401454571546123640940296362684428021715810487951657759510487903014385010799379039019523703113225888923253062427006216845993174719967130651564620845163319
144782969330614053664844635165876817564392295340867154999705446242937148622958602769384780126584244964331371382820007110827243511371606614303665422874073139595209840071135419036045361039858229395
Technical details
Node id
n-9d0af41400c21a44 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b373e0a412d5b427

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:     3,      4;
  2:     9,     25,      17;
  3:    27,    118,     167,       77;
  4:    81,    499,    1108,     1061,      372;
  5:   243,   1996,    6214,     9301,     6754,     1915;
  6:   729,   7741,   31851,    66464,    75141,    43996,    10481;
  7:  2187,  29506,  154881,   423781,   664439,   603422,   296015,    60814;
  8:  6561, 111463,  729590,  2520531,  5148011,  6454447,  4894772,  2065431,   372939;
  9: 19683, 419608, 3372536, 14342945, 36699599, 59624965, 62187586, 40449145, 14967095, 2409837;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3-(n-8)*x-(2*n-6)*x^2-(n-1)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx 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[((3 - (n - 8)*x - (2*n - 6)*x^2 - (n - 1)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + x)]];
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] = ((3 - (n - 8)*x - (2*n - 6)*x^2 - (n - 1)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + x)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.