Real-rooted atlas

A013609 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 12\,x + 9\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13912
295481144
3272437297291728
48197243748748656120736
5243364521870656109841559049248832
6729131229841539366088573510628825314412985984
7218745927413343206671562001451116026111160261478296935831808
8656115746416533729920232372008708928208813392313211479125643046721429981696
9196835314416377292446410442008846986026540941205308188154968195611622614673874204895159780352
1059049177147023914845191318760100442349036159245649039811410154968195601743392200511622614670348678440161917364224
111771475845851876877657891898854735139310198875851025966275530612784876137019177314205519177314205511506388523331381059609743008370688
125314411913187631567595431567595402130812689510227900909635797653183692051108186417259582784952301277704660207114993419411297181459242824295364818916100448256
13159432362178597111921474612311362206923352165454986101693431994440677372598332203211613462474572261224374576204352692494914452222029503845518110147519227592541865828329106993205379072
1447829692008846983917251611470070193323878079094892326847456934104708135562033589993219269694237322005827188474644011654282711966017481308413053837252231309790377939106758364789818228767924549611283918464548864
Technical details
Node id
n-390b8fb6037369c6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f9762973ec1089d7

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     3,      9;
  2:     9,     54,      81;
  3:    27,    243,     729,      729;
  4:    81,    972,    4374,     8748,      6561;
  5:   243,   3645,   21870,    65610,     98415,     59049;
  6:   729,  13122,   98415,   393660,    885735,   1062882,     531441;
  7:  2187,  45927,  413343,  2066715,   6200145,  11160261,   11160261,    4782969;
  8:  6561, 157464, 1653372,  9920232,  37200870,  89282088,  133923132,  114791256,   43046721;
  9: 19683, 531441, 6377292, 44641044, 200884698, 602654094, 1205308188, 1549681956, 1162261467, 387420489;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3+12*x+9*x^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[((3 + 12*x + 9*x^2)*P[n - 1, 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 + 12*x + 9*x^2)*v[i - 1])/(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

Candidate equalities ?

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.