Real-rooted atlas

A345013 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 6\,x + 2\,x^{2} - 6\,x^{3} - 4\,x^{4} + 6\,n + 22\,n x + 26\,n x^{2} + 10\,n x^{3} + 4\,n^{2} + 17\,n^{2} x + 26\,n^{2} x^{2} + 17\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 10\,x^{2} - 18\,x^{3} - 14\,x^{4} - 4\,x^{5} - 2\,n x - 10\,n x^{2} - 18\,n x^{3} - 14\,n x^{4} - 4\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14711
2155041106
356273438231998
421013443192333212899367
57926270196803060023560718388085
630032831411038522770026203515946240081830980
71144012512558258214964952289980208708510488102241437865660
843758544544294694490570481728727020980960158096406761480125646574688109
916796023430421444809651682176118171872179090730179874240115441560429550207059735711234431
10646646999362069157530282172800751656360136579060817142106801467331320819724815269860890397532736790298542
112496144423304423248152601488529350452623776095880689521443709713615450860040115171980005694384465168062710222429823164976942882
1296577001782960001502322096763909212026105703030631609704001109286393601424864422081328416471808766248832038864921040103934561401267854873623041490467
13374421607475059806860830080383294797601454140573603955819030027938403661761189787254656133175302603210994654041806507152995202614397292006390871656071784612155985059475881
1414542267531217400903099976834518873656802078703446321523777981727505367145957173919430732906412011544183639119083113529388818962997845473066402718017375044254753911028069904070477878557588083994184
Technical details
Node id
n-cdeac028c94ac988 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-7301b23e04f602de

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:      4,       7;
  2:     15,      50,       41;
  3:     56,     273,      438,      231;
  4:    210,    1344,     3192,     3332,      1289;
  5:    792,    6270,    19680,    30600,     23560,      7183;
  6:   3003,   28314,   110385,   227700,    262035,    159462,     40081;
  7:  11440,  125125,   582582,  1496495,   2289980,   2087085,   1048810,    224143;
  8:  43758,  544544,  2946944,  9057048,  17287270,  20980960,  15809640,   6761480,  1256465;
  9: 167960, 2343042, 14448096, 51682176, 118171872, 179090730, 179874240, 115441560, 42955020, 7059735;
%F Formula
The row polynomials satisfy (n^2+2*n+(2*n^2+4*n)*x+(n^2+2*n)*x^2)*P(n,x) = (4*n^2+6*n+2+(17*n^2+22*n+6)*x+(26*n^2+26*n+2)*x^2+(17*n^2+10*n-6)*x^3+(4*n^2-4)*x^4)*P(n-1,x) - ((2*n+2)*x+(10*n+10)*x^2+(18*n+18)*x^3+(14*n+14)*x^4+(4*n+4)*x^5)*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[((4*n^2 + 6*n + 2 + (17*n^2 + 22*n + 6)*x + (26*n^2 + 26*n + 2)*x^2 + (17*n^2 + 10*n - 6)*x^3 + (4*n^2 - 4)*x^4)*P[n - 1, x] - ((2*n + 2)*x + (10*n + 10)*x^2 + (18*n + 18)*x^3 + (14*n + 14)*x^4 + (4*n + 4)*x^5)*D[P[n - 1, x], x])/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*x^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*n^2 + 6*n + 2 + (17*n^2 + 22*n + 6)*x + (26*n^2 + 26*n + 2)*x^2 + (17*n^2 + 10*n - 6)*x^3 + (4*n^2 - 4)*x^4)*v[i - 1] - ((2*n + 2)*x + (10*n + 10)*x^2 + (18*n + 18)*x^3 + (14*n + 14)*x^4 + (4*n + 4)*x^5)*deriv(v[i - 1], x))/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*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

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.