Real-rooted atlas

A113214 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(30\,n + 81\,n x + 27\,n x^{2} + 45\,n^{2} x + 54\,n^{2} x^{2} + 27\,n^{3} x^{2}\right)\,P_{n}(x) = \left(150 + 450\,x + 486\,x^{2} + 162\,x^{3} - 60\,n + 243\,n x + 621\,n x^{2} + 324\,n x^{3} - 135\,n^{2} x + 27\,n^{2} x^{2} + 162\,n^{2} x^{3} - 54\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(25 + 150\,x + 306\,x^{2} + 270\,x^{3} + 81\,x^{4} + 75\,n x + 360\,n x^{2} + 459\,n x^{3} + 162\,n x^{4} + 45\,n^{2} x^{2} + 162\,n^{2} x^{3} + 81\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
24302761
3778189108382
4111747029724052264
518360202546444455145812960
6297085130165242592018954510372368
747135011961497881097551297627654517496396704
876251426352134676384345637146602154297432590492143744
91234602557013377161185840254566833628772641896112193119683011453184
10199831011407580060433416558835480151048801653372011101212413343064953960623104
1132214844227880181742487961952768013058301775824936407694084745074070149393972125764318412288
1252126280446148398584821955455802118702012112903507073204222593993426810301779736865314410069087331661511680
13843461828590868499384525139202185979406371350651322960040196125348620517559201472504913686621772186535791223205228621604864
14136480646163131317706168121288995566899560188418069045479977208032140225103394799009552120534614133219625976836086472951387174453544521582592
Technical details
Node id
n-ef954f7282052a0e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b0d276e27b0dc736

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:   3,    6;
  2:   4,   30,    27;
  3:   7,   78,   189,    108;
  4:  11,  174,   702,    972,     405;
  5:  18,  360,  2025,   4644,    4455,    1458;
  6:  29,  708,  5130,  16524,   25920,   18954,    5103;
  7:  47, 1350, 11961,  49788,  109755,  129762,   76545,   17496;
  8:  76, 2514, 26352, 134676,  384345,  637146,  602154,  297432,   59049;
  9: 123, 4602, 55701, 337716, 1185840, 2545668, 3362877, 2641896, 1121931, 196830;
%F Formula
The row polynomials satisfy (30*n+(45*n^2+81*n)*x+(27*n^3+54*n^2+27*n)*x^2)*P(n,x) = -(60*n-150+(135*n^2-243*n-450)*x+(54*n^3-27*n^2-621*n-486)*x^2-(162*n^2+324*n+162)*x^3)*P(n-1,x) + (25+(75*n+150)*x+(45*n^2+360*n+306)*x^2+(162*n^2+459*n+270)*x^3+(81*n^2+162*n+81)*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[(-(60*n - 150 + (135*n^2 - 243*n - 450)*x + (54*n^3 - 27*n^2 - 621*n - 486)*x^2 - (162*n^2 + 324*n + 162)*x^3)*P[n - 1, x] + (25 + (75*n + 150)*x + (45*n^2 + 360*n + 306)*x^2 + (162*n^2 + 459*n + 270)*x^3 + (81*n^2 + 162*n + 81)*x^4)*D[P[n - 1, x], x])/(30*n + (45*n^2 + 81*n)*x + (27*n^3 + 54*n^2 + 27*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] = (-(60*n - 150 + (135*n^2 - 243*n - 450)*x + (54*n^3 - 27*n^2 - 621*n - 486)*x^2 - (162*n^2 + 324*n + 162)*x^3)*v[i - 1] + (25 + (75*n + 150)*x + (45*n^2 + 360*n + 306)*x^2 + (162*n^2 + 459*n + 270)*x^3 + (81*n^2 + 162*n + 81)*x^4)*deriv(v[i - 1], x))/(30*n + (45*n^2 + 81*n)*x + (27*n^3 + 54*n^2 + 27*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.