Real-rooted atlas

A102537 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2} + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-4\,x - 18\,x^{2} - 24\,x^{3} - 10\,x^{4} + 2\,n + 18\,n x + 69\,n x^{2} + 92\,n x^{3} + 39\,n x^{4} + 4\,n^{2} + 22\,n^{2} x + 15\,n^{2} x^{2} - 20\,n^{2} x^{3} - 17\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 11\,x^{2} - 21\,x^{3} - 17\,x^{4} - 5\,x^{5} + 8\,n x + 41\,n x^{2} + 75\,n x^{3} + 59\,n x^{4} + 17\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2145
31102132
411888126233
51282347528181833
614050026706375559415180
71549357300284435396639693130392
8170159616940950602906404572402895101151057
9188254835056264040115024028890403880720215715010378883
10110838646661264323037261561322740828164744330013621634896095182445
111130562511844014193901045573249006230146551200270660690281193940125642146885053524
12115479201996502897796263007361557660726115741501578304035257264055424004619329767896208324942620
1311801084632208055548906066878444020284021728971447328841685166234879802424294767420527817160766846596479071217228
141208145085007861010652513040141411327521926818898372287952569198503931150017197697488422686949888217582891766760708178054757310811912
1512381901975457217594577264249986269921952319371845064993867744873662151392909612037364931753178416444211102546172715082715584864840939139177305728683824
Technical details
Node id
n-614d2749fe4b2963 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-84e6bc5890e6c39e

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,   4;
   3: 1,  10,   21;
   4: 1,  18,   88,   126;
   5: 1,  28,  234,   752,    818;
   6: 1,  40,  500,  2670,   6375,    5594;
   7: 1,  54,  935,  7300,  28443,   53966,    39693;
   8: 1,  70, 1596, 16940,  95060,  290640,   457240,   289510;
   9: 1,  88, 2548, 35056, 264040, 1150240,  2889040,  3880720,  2157150;
  10: 1, 108, 3864, 66612, 643230, 3726156, 13227408, 28164744, 33001362, 16348960;
%F Formula
The row polynomials satisfy (4*n^2+2*n+(8*n^2+4*n)*x+(4*n^2+2*n)*x^2)*P(n,x) = (4*n^2+2*n+(22*n^2+18*n-4)*x+(15*n^2+69*n-18)*x^2-(20*n^2-92*n+24)*x^3-(17*n^2-39*n+10)*x^4)*P(n-1,x) + ((8*n-2)*x+(41*n-11)*x^2+(75*n-21)*x^3+(59*n-17)*x^4+(17*n-5)*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 2*n + (22*n^2 + 18*n - 4)*x + (15*n^2 + 69*n - 18)*x^2 - (20*n^2 - 92*n + 24)*x^3 - (17*n^2 - 39*n + 10)*x^4)*P[n - 1, x] + ((8*n - 2)*x + (41*n - 11)*x^2 + (75*n - 21)*x^3 + (59*n - 17)*x^4 + (17*n - 5)*x^5)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (8*n^2 + 4*n)*x + (4*n^2 + 2*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n^2 + 2*n + (22*n^2 + 18*n - 4)*x + (15*n^2 + 69*n - 18)*x^2 - (20*n^2 - 92*n + 24)*x^3 - (17*n^2 - 39*n + 10)*x^4)*v[i - 1] + ((8*n - 2)*x + (41*n - 11)*x^2 + (75*n - 21)*x^3 + (59*n - 17)*x^4 + (17*n - 5)*x^5)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (8*n^2 + 4*n)*x + (4*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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.