Real-rooted atlas

A143494 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(4\,x - n - n^{2} - 7\,n^{2} x^{2} - 4\,n^{3} x - 2\,n^{3} x^{2} + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-x + 5\,x^{2} + 4\,n^{2} x^{2} - 13\,n^{2} x^{3} + 4\,n^{3} x^{2} - 14\,n^{3} x^{3} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(x^{3} - 3\,n^{2} x^{4} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(n + 2\,n^{2} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(x - 4\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2189
31203657
4136171128336
515649510404001992
6180112545605275115212193
711082205145603377523940313677725
811403906380801475252184841008918192517219
9117664268668850452512806641290415403520207363593151
10121699901780801452675563976010006045714528015528515120026036098
111260148503379203678675202827245606408572490880377258315802500123904196521630
12130821285601920843067562661060249313372511850240496457181192102900211869792949121542920833
131360296011018160178356751719226089316226922791286784438331994132566534009509650157593768069222412581284141
1414164013116496483533530042875632830376746401252579328029112376293357256900002065371808546028455202680561391605632106393541413
15147653235257712066266200988467480887352466048233968640155936033253287314127100280022227485127501002720218758587199342506603686400931752044149
Technical details
Node id
n-8c583e5c6c2587ca · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4ac9254ff888a6a8

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,   8;
   3: 1,  20,   36;
   4: 1,  36,  171,    128;
   5: 1,  56,  495,   1040,     400;
   6: 1,  80, 1125,   4560,    5275,    1152;
   7: 1, 108, 2205,  14560,   33775,   23940,     3136;
   8: 1, 140, 3906,  38080,  147525,  218484,   100891,    8192;
   9: 1, 176, 6426,  86688,  504525, 1280664,  1290415,  403520,   20736;
  10: 1, 216, 9990, 178080, 1452675, 5639760, 10006045, 7145280, 1552851, 51200;
%F Formula
The row polynomials satisfy n^2*P(n,x) = -(n^2+n+(4*n^3-4)*x-(9*n^4-2*n^3-7*n^2)*x^2)*P(n-2,x) - (x-(4*n^3+4*n^2+5)*x^2-(4*n^4-14*n^3-13*n^2)*x^3)*d/dx P(n-2,x) + (x^3-(4*n^3+3*n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^2+n+4*n^3*x)*P(n-1,x) + (x-(4*n^3+4*n^2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 8*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + n + (4*n^3 - 4)*x - (9*n^4 - 2*n^3 - 7*n^2)*x^2)*P[n - 2, x] - (x - (4*n^3 + 4*n^2 + 5)*x^2 - (4*n^4 - 14*n^3 - 13*n^2)*x^3)*D[P[n - 2, x], x] + (x^3 - (4*n^3 + 3*n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 + n + 4*n^3*x)*P[n - 1, x] + (x - (4*n^3 + 4*n^2)*x^2)*D[P[n - 1, x], x])/(n^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 8*x; for(n = 3, N, my(i = n); v[i] = (-(n^2 + n + (4*n^3 - 4)*x - (9*n^4 - 2*n^3 - 7*n^2)*x^2)*v[i - 2] - (x - (4*n^3 + 4*n^2 + 5)*x^2 - (4*n^4 - 14*n^3 - 13*n^2)*x^3)*deriv(v[i - 2], x) + (x^3 - (4*n^3 + 3*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + n + 4*n^3*x)*v[i - 1] + (x - (4*n^3 + 4*n^2)*x^2)*deriv(v[i - 1], x))/(n^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. [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.