Real-rooted atlas

A113953 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(18\,n + 23\,n x + 5\,n x^{2} + 9\,n^{2} x + 7\,n^{2} x^{2} + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(90 + 142\,x + 62\,x^{2} + 10\,x^{3} - 72\,n - 20\,n x + 42\,n x^{2} + 14\,n x^{3} - 54\,n^{2} x - 18\,n^{2} x^{2} + 4\,n^{2} x^{3} - 8\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(81 + 216\,x + 194\,x^{2} + 64\,x^{3} + 5\,x^{4} + 81\,n x + 153\,n x^{2} + 79\,n x^{3} + 7\,n x^{4} + 18\,n^{2} x^{2} + 20\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
238314
352221452
4116084405200
521146282216656736
6433528498964459672688
78581423793212223579813389664

All 15 rows

Technical details
Node id
n-8f99ad5fd195ac4d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7769c1967f67c497

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   1,    2;
  2:   3,    8,     3;
  3:   5,   22,    21,     4;
  4:  11,   60,    84,    40,     5;
  5:  21,  146,   282,   216,    65,     6;
  6:  43,  352,   849,   896,   445,    96,    7;
  7:  85,  814,  2379,  3212,  2235,   798,  133,    8;
  8: 171, 1860,  6354, 10440,  9380,  4740, 1302,  176,   9;
  9: 341, 4170, 16344, 31680, 34930, 22956, 8960, 1984, 225, 10;
%F Formula
The row polynomials satisfy (18*n+(9*n^2+23*n)*x+(2*n^3+7*n^2+5*n)*x^2)*P(n,x) = -(72*n-90+(54*n^2+20*n-142)*x+(8*n^3+18*n^2-42*n-62)*x^2-(4*n^2+14*n+10)*x^3)*P(n-1,x) + (81+(81*n+216)*x+(18*n^2+153*n+194)*x^2+(20*n^2+79*n+64)*x^3+(2*n^2+7*n+5)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(72*n - 90 + (54*n^2 + 20*n - 142)*x + (8*n^3 + 18*n^2 - 42*n - 62)*x^2 - (4*n^2 + 14*n + 10)*x^3)*P[n - 1, x] + (81 + (81*n + 216)*x + (18*n^2 + 153*n + 194)*x^2 + (20*n^2 + 79*n + 64)*x^3 + (2*n^2 + 7*n + 5)*x^4)*D[P[n - 1, x], x])/(18*n + (9*n^2 + 23*n)*x + (2*n^3 + 7*n^2 + 5*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(72*n - 90 + (54*n^2 + 20*n - 142)*x + (8*n^3 + 18*n^2 - 42*n - 62)*x^2 - (4*n^2 + 14*n + 10)*x^3)*v[i - 1] + (81 + (81*n + 216)*x + (18*n^2 + 153*n + 194)*x^2 + (20*n^2 + 79*n + 64)*x^3 + (2*n^2 + 7*n + 5)*x^4)*deriv(v[i - 1], x))/(18*n + (9*n^2 + 23*n)*x + (2*n^3 + 7*n^2 + 5*n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.