Real-rooted atlas

A094305 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(8 + 60\,x + 186\,x^{2} + 305\,x^{3} + 279\,x^{4} + 135\,x^{5} + 27\,x^{6}\right)\,P_{n-3}(x) + \left(-12 - 72\,x - 171\,x^{2} - 201\,x^{3} - 117\,x^{4} - 27\,x^{5}\right)\,P_{n-2}(x) + \left(6 + 27\,x + 45\,x^{2} + 33\,x^{3} + 9\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 9\,x\)
\(P_{2}(x) = 24 + 72\,x + 54\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16915
2247254150
3803605402701250
424014403240324012159375
56725040151202268017010510365625
6179216128604801209601360808164820412437500
7460848384217728544320816480734832367416787322812500
8115201382407257602177280408240048988803674160157464029524517578125
928160380160228096079833601796256026943840269438401732104064953901082565107421875
106758410137606842880273715207185024012933043216166304013856832077944680259815603897234644531250
1115974426357761976832088957440266872320560431872840647808900694080675520560337760280101328084138174663808593750
123727366709248553512962767564809340531202241727488392302310450438868484728643920315242928014185931763868890484836113122216796875
138601601677312015095808083026944031135104008406478080168129561602521943424028371863520236432196001418593176058033357201450833930167403915128173828125
1419660804128768040255488024153292809963233280298896998406725182464011528884224015131660544015131660544011348745408061902247680232133428805356925280573956280732421875000
Technical details
Node id
n-10e8604b45f3a91d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f06e51c139cfae05

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:     6,      9;
  2:    24,     72,      54;
  3:    80,    360,     540,     270;
  4:   240,   1440,    3240,    3240,     1215;
  5:   672,   5040,   15120,   22680,    17010,     5103;
  6:  1792,  16128,   60480,  120960,   136080,    81648,    20412;
  7:  4608,  48384,  217728,  544320,   816480,   734832,   367416,    78732;
  8: 11520, 138240,  725760, 2177280,  4082400,  4898880,  3674160,  1574640,  295245;
  9: 28160, 380160, 2280960, 7983360, 17962560, 26943840, 26943840, 17321040, 6495390, 1082565;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (8+60*x+186*x^2+305*x^3+279*x^4+135*x^5+27*x^6)*P(n-3,x) - (12+72*x+171*x^2+201*x^3+117*x^4+27*x^5)*P(n-2,x) + (6+27*x+45*x^2+33*x^3+9*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 9*x, P(2,x) = 24 + 72*x + 54*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 9*x;
P[2, x_] := 24 + 72*x + 54*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8 + 60*x + 186*x^2 + 305*x^3 + 279*x^4 + 135*x^5 + 27*x^6)*P[n - 3, x] - (12 + 72*x + 171*x^2 + 201*x^3 + 117*x^4 + 27*x^5)*P[n - 2, x] + (6 + 27*x + 45*x^2 + 33*x^3 + 9*x^4)*P[n - 1, x])/(1 + 3*x + 3*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 9*x; v[3] = 24 + 72*x + 54*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((8 + 60*x + 186*x^2 + 305*x^3 + 279*x^4 + 135*x^5 + 27*x^6)*v[i - 3] - (12 + 72*x + 171*x^2 + 201*x^3 + 117*x^4 + 27*x^5)*v[i - 2] + (6 + 27*x + 45*x^2 + 33*x^3 + 9*x^4)*v[i - 1])/(1 + 3*x + 3*x^2 + x^3)); 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. [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.