Real-rooted atlas

A331333 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,x\right)\,P_{n}(x) = \left(3 + 17\,x + 23\,x^{2} + 11\,x^{3} + 2\,x^{4} - n - 10\,n x - 20\,n x^{2} - 14\,n x^{3} - 3\,n x^{4} + n^{2} x + 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}(x) + \left(1 + 12\,x + 34\,x^{2} + 40\,x^{3} + 21\,x^{4} + 4\,x^{5} - 2\,n x - 8\,n x^{2} - 12\,n x^{3} - 8\,n x^{4} - 2\,n x^{5}\right)\,P_{n-1}'(x) + \left(x + 5\,x^{2} + 10\,x^{3} + 10\,x^{4} + 5\,x^{5} + x^{6}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
22121428
365412686272
424288100813766483344
5120180084001720016200575249472
6720129607560020640029160020707258576852928
75040105840740880252840047628005073264287022467156816758016
8403209676807902720323635207620480010822963291847168429803528546432369082624
936288097977609144576043690752012345177602191650048247987353617407042566922609921194018568996922368
103628800108864000114307200062415360002057529600043833000960619968384005802347520034613049600119401856001815177984240294124544
113991680013172544001536796800094403232000355658688000883965519360150032348928017552101248001396059667200722381228800219636536064298089080326974172532736
1247900160017244057600221298739200151045171200064018563840001818443354112036007763742720505500515942405025814801920034674298982400158138305966084292482756608525586164736218457925292032
1362270208002428538112003399953356800255266339328001202126365440003841461585561608693302960742401423826453237760169872540304896014649891320064008908457902755843627147929333768882406184038498983436912647342688736329728
14871782912003661488230400555325714944004548382046208002356167676262400836584967522304021298592253818880398671406906572805549169649959936057427573974650880436514437235036162369736647164723287047580603576321940075363487744198227905206272263513011737051136
Technical details
Node id
n-e4b9660d23c860b7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4f9e606a0e083ccf

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:      1,       3;
  2:      2,      12,       14;
  3:      6,      54,      126,        86;
  4:     24,     288,     1008,      1376,        648;
  5:    120,    1800,     8400,     17200,      16200,       5752;
  6:    720,   12960,    75600,    206400,     291600,     207072,      58576;
  7:   5040,  105840,   740880,   2528400,    4762800,    5073264,    2870224,     671568;
  8:  40320,  967680,  7902720,  32363520,   76204800,  108229632,   91847168,   42980352,   8546432;
  9: 362880, 9797760, 91445760, 436907520, 1234517760, 2191650048, 2479873536, 1740704256, 692260992, 119401856;
%F Formula
The row polynomials satisfy (2+2*x)*P(n,x) = -(n-3-(n^2-10*n+17)*x-(3*n^2-20*n+23)*x^2-(3*n^2-14*n+11)*x^3-(n^2-3*n+2)*x^4)*P(n-1,x) + (1-(2*n-12)*x-(8*n-34)*x^2-(12*n-40)*x^3-(8*n-21)*x^4-(2*n-4)*x^5)*d/dx P(n-1,x) + (x+5*x^2+10*x^3+10*x^4+5*x^5+x^6)*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[(-(n - 3 - (n^2 - 10*n + 17)*x - (3*n^2 - 20*n + 23)*x^2 - (3*n^2 - 14*n + 11)*x^3 - (n^2 - 3*n + 2)*x^4)*P[n - 1, x] + (1 - (2*n - 12)*x - (8*n - 34)*x^2 - (12*n - 40)*x^3 - (8*n - 21)*x^4 - (2*n - 4)*x^5)*D[P[n - 1, x], x] + (x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6)*D[P[n - 1, x], {x, 2}])/(2 + 2*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] = (-(n - 3 - (n^2 - 10*n + 17)*x - (3*n^2 - 20*n + 23)*x^2 - (3*n^2 - 14*n + 11)*x^3 - (n^2 - 3*n + 2)*x^4)*v[i - 1] + (1 - (2*n - 12)*x - (8*n - 34)*x^2 - (12*n - 40)*x^3 - (8*n - 21)*x^4 - (2*n - 4)*x^5)*deriv(v[i - 1], x) + (x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6)*deriv(deriv(v[i - 1], x), x))/(2 + 2*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

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.