Real-rooted atlas

A048854 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 14\,x + 80\,x^{2} + 8\,n x - 72\,n x^{2} + 16\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-8\,x^{2} + 88\,x^{3} - 32\,n x^{3}\right)\,P_{n-1}'(x) + \left(16\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2123
31121225
4130180120331
5156840336016805937
61902520252007560030240133651
71132594011088083160019958406652803609673
81182120123603605045040302702406054048017297280113525595
912402184096096021621600242161920121080960020756736005189184004070168161
10130636720222768073513440132324192012350257920529296768007939451520017643225600163716695587
111380581404651200211629600558702144083805321600670442572800251415964800033522128640006704425728007296866339961
1214628778089535605372136001955457504043020065088055311512256003871805857920012906019526400015487223431680028158588057600356790528924523
13155212751216151520123559128059308381440179902090368033924394183680381649434566400237470759285760071241227785728007771770303897600129529505064960018983329135883665
1416501794002762760026246220001606268664006425074656000167051941056000275635702742400027563570274240000154355993535744000420970891461120000420970891461120000647647525324800001091556096587136051
15175624570045208800522161640039684284640020238985166400693908062848000157864084297920002315339903036160002083805912732544000106084664648202240002652116616205056000024481076457277440000349729663675392000067439846581526028457
Technical details
Node id
n-a50309a7fde94c6d · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-14fed6d19df1fb72

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,   2;
   3: 1,  12,    12;
   4: 1,  30,   180,     120;
   5: 1,  56,   840,    3360,     1680;
   6: 1,  90,  2520,   25200,    75600,      30240;
   7: 1, 132,  5940,  110880,   831600,    1995840,      665280;
   8: 1, 182, 12012,  360360,  5045040,   30270240,    60540480,    17297280;
   9: 1, 240, 21840,  960960, 21621600,  242161920,  1210809600,  2075673600,   518918400;
  10: 1, 306, 36720, 2227680, 73513440, 1323241920, 12350257920, 52929676800, 79394515200, 17643225600;
%F Formula
The row polynomials satisfy P(n,x) = (1+(8*n-14)*x+(16*n^2-72*n+80)*x^2)*P(n-1,x) - (8*x^2+(32*n-88)*x^3)*d/dx P(n-1,x) + 16*x^4*d^2/dx^2 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[(1 + (8*n - 14)*x + (16*n^2 - 72*n + 80)*x^2)*P[n - 1, x] - (8*x^2 + (32*n - 88)*x^3)*D[P[n - 1, x], x] + 16*x^4*D[P[n - 1, x], {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] = (1 + (8*n - 14)*x + (16*n^2 - 72*n + 80)*x^2)*v[i - 1] - (8*x^2 + (32*n - 88)*x^3)*deriv(v[i - 1], x) + 16*x^4*deriv(deriv(v[i - 1], x), x)); 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.