Real-rooted atlas

A143494 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,x + n\right)\,P_{n}(x) = \left(-2\,x + 8\,x^{2} - 6\,x^{3} + 2\,x^{4} - n + 4\,n x - 7\,n x^{2} + 7\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x + 3\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x^{3} - 2\,x^{4} + 2\,x^{5} + n x^{2} - n x^{3} + 3\,n x^{4} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-1 + 6\,x - 6\,x^{2} + 2\,x^{3} + 2\,n - 4\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(x^{2} - 2\,x^{3} - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2325
3613423
41047438108
51512425012916534
6212709801129371322803
72851729756314467610496415623
8369037574259843624618423295512892249
94514721696886142198051193584704708353256575341
10552274345002435588414071381890985720265013237795123778708
1166336565010609708296419274283999072855485867898797668265102426059538
12784807115225138652890401523226875361071648570304682341836236720191976272048188207715
139166681941942917343245915671188518763243097004765343443472138671111586841366001057649740961420018937
1410590223137685756751609629023838700581432002000307101194435747241532064175750522298140509977771692627819211167823189
1512011949489125107636621399377981114011470545668266016154745464277834076522601547412112053893215243873904219140855649959611638491364577179
Technical details
Node id
n-9d39155058a4737c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ac6555d98267de92

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:  3,    2;
   3:  6,   13,     4;
   4: 10,   47,    43,      8;
   5: 15,  124,   250,    129,     16;
   6: 21,  270,   980,   1129,    371,      32;
   7: 28,  517,  2975,   6314,   4676,    1049,     64;
   8: 36,  903,  7574,  25984,  36246,   18423,   2955,    128;
   9: 45, 1472, 16968,  86142, 198051,  193584,  70470,   8353,   256;
  10: 55, 2274, 34500, 243558, 841407, 1381890, 985720, 265013, 23779, 512;
%F Formula
The row polynomials satisfy (n-1+2*x)*P(n,x) = -(n+(2*n^2-4*n+2)*x+(n^3-3*n^2+7*n-8)*x^2+(3*n^2-7*n+6)*x^3+(2*n-2)*x^4)*P(n-2,x) + (n*x^2+(n^2-n+2)*x^3+(3*n-2)*x^4+2*x^5)*d/dx P(n-2,x) + (2*n-1+(2*n^2-4*n+6)*x+(5*n-6)*x^2+2*x^3)*P(n-1,x) - ((n-1)*x^2+2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + (2*n^2 - 4*n + 2)*x + (n^3 - 3*n^2 + 7*n - 8)*x^2 + (3*n^2 - 7*n + 6)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] + (n*x^2 + (n^2 - n + 2)*x^3 + (3*n - 2)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n - 1 + (2*n^2 - 4*n + 6)*x + (5*n - 6)*x^2 + 2*x^3)*P[n - 1, x] - ((n - 1)*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n - 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(n + (2*n^2 - 4*n + 2)*x + (n^3 - 3*n^2 + 7*n - 8)*x^2 + (3*n^2 - 7*n + 6)*x^3 + (2*n - 2)*x^4)*v[i - 2] + (n*x^2 + (n^2 - n + 2)*x^3 + (3*n - 2)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (2*n - 1 + (2*n^2 - 4*n + 6)*x + (5*n - 6)*x^2 + 2*x^3)*v[i - 1] - ((n - 1)*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n - 1 + 2*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. [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. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.