Real-rooted atlas

A102537 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(2\,n - 2\,n x + 2\,n^{2} + 4\,n^{2} x - 4\,n^{3} + 6\,n^{3} x - 16\,n^{4} x + 8\,n^{5} x\right)\,P_{n}(x) = \left(2\,n - 2\,n x + 2\,n^{2} + 16\,n^{2} x - 4\,n^{3} - 30\,n^{3} x + 6\,n^{3} x^{2} - 16\,n^{4} x - 30\,n^{4} x^{2} + 8\,n^{5} x + 36\,n^{5} x^{2}\right)\,P_{n-1}(x) + \left(x - x^{2} + n x + 8\,n x^{2} - 8\,n^{2} x - 15\,n^{2} x^{2} + 3\,n^{2} x^{3} - 8\,n^{3} x^{2} - 15\,n^{3} x^{3} + 16\,n^{4} x^{2} + 18\,n^{4} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
211213
3132108141
41604958801436
519614045824682514150
611403150224005950051408136599
711926120652802907005581443798481300285
8125210773159600104737533180844946403276883212251320
9132017640344960309925014281344347116004209920019980675114534990
101396273246800647969500497296801740538803410035203476637451430715001064199610
1114804050012480001842750014859936069827940019543680003194933625280488000010181714409838948306
1215725791521621603918915039502663223811327548885859840206179716602884819080022213106916720966355290592361952
1316728038835728007787587595693875271637498243408524697610485207810020711521600025288767873617327840947250840366668831261214264
141780108810567424014628900021486928321949739792011459695104044719669566011655332260002000055015816216384614784013348726814803572622379207605161119339
15189614414487127042620618004528427904488692844643465017487361662802337196547424637760012344425581488186974920903681814565638631210175443180672250307002440069403306358685
Technical details
Node id
n-9be10ad9a53be547 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c4cc0e4be31b0229

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,  12;
   3: 1,  32,   108;
   4: 1,  60,   495,    880;
   5: 1,  96,  1404,   5824,    6825;
   6: 1, 140,  3150,  22400,   59500,    51408;
   7: 1, 192,  6120,  65280,  290700,   558144,    379848;
   8: 1, 252, 10773, 159600, 1047375,  3318084,   4946403,   2768832;
   9: 1, 320, 17640, 344960, 3099250, 14281344,  34711600,  42099200,  19980675;
  10: 1, 396, 27324, 680064, 7969500, 49729680, 174053880, 341003520, 347663745, 143071500;
%F Formula
The row polynomials satisfy -(4*n^3-2*n^2-2*n-(8*n^5-16*n^4+6*n^3+4*n^2-2*n)*x)*P(n,x) = -(4*n^3-2*n^2-2*n-(8*n^5-16*n^4-30*n^3+16*n^2-2*n)*x-(36*n^5-30*n^4+6*n^3)*x^2)*P(n-1,x) - ((8*n^2-n-1)*x-(16*n^4-8*n^3-15*n^2+8*n-1)*x^2-(18*n^4-15*n^3+3*n^2)*x^3)*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^3 - 2*n^2 - 2*n - (8*n^5 - 16*n^4 - 30*n^3 + 16*n^2 - 2*n)*x - (36*n^5 - 30*n^4 + 6*n^3)*x^2)*P[n - 1, x] - ((8*n^2 - n - 1)*x - (16*n^4 - 8*n^3 - 15*n^2 + 8*n - 1)*x^2 - (18*n^4 - 15*n^3 + 3*n^2)*x^3)*D[P[n - 1, x], x])/(-4*n^3 + 2*n^2 + 2*n + (8*n^5 - 16*n^4 + 6*n^3 + 4*n^2 - 2*n)*x)]];
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^3 - 2*n^2 - 2*n - (8*n^5 - 16*n^4 - 30*n^3 + 16*n^2 - 2*n)*x - (36*n^5 - 30*n^4 + 6*n^3)*x^2)*v[i - 1] - ((8*n^2 - n - 1)*x - (16*n^4 - 8*n^3 - 15*n^2 + 8*n - 1)*x^2 - (18*n^4 - 15*n^3 + 3*n^2)*x^3)*deriv(v[i - 1], x))/(-4*n^3 + 2*n^2 + 2*n + (8*n^5 - 16*n^4 + 6*n^3 + 4*n^2 - 2*n)*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

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.