Real-rooted atlas

A030528 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(30 + 72\,x + 30\,n + 117\,n x + 27\,n x^{2} - 81\,n x^{3} + 45\,n^{2} x + 54\,n^{2} x^{2} - 81\,n^{2} x^{3} + 27\,n^{3} x^{2}\right)\,P_{n}(x) = \left(180 + 612\,x + 594\,x^{2} - 324\,x^{3} - 486\,x^{4} - 60\,n + 351\,n x + 972\,n x^{2} + 243\,n x^{3} - 486\,n x^{4} - 135\,n^{2} x + 108\,n^{2} x^{2} + 405\,n^{2} x^{3} - 54\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(25 + 135\,x + 162\,x^{2} - 216\,x^{3} - 567\,x^{4} - 243\,x^{5} + 75\,n x + 315\,n x^{2} + 162\,n x^{3} - 405\,n x^{4} - 243\,n x^{5} + 45\,n^{2} x^{2} + 162\,n^{2} x^{3} + 81\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1268
23242754
3560162108335
481325678644051976
513270159338884050145811272
6215283969133922227517496510362784
7341002915339528911251137247144217496343504

All 15 rows

Technical details
Node id
n-83cc2466332c48a7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-a99eed42e26d3f2e

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:  2,    6;
  2:  3,   24,    27;
  3:  5,   60,   162,    108;
  4:  8,  132,   567,    864,    405;
  5: 13,  270,  1593,   3888,   4050,    1458;
  6: 21,  528,  3969,  13392,  22275,   17496,    5103;
  7: 34, 1002,  9153,  39528,  91125,  113724,   71442,   17496;
  8: 55, 1860, 20007, 105408, 311850,  539460,  535815,  279936,   59049;
  9: 89, 3396, 42039, 261576, 946485, 2103894, 2893401, 2379456, 1062882, 196830;
%F Formula
The row polynomials satisfy (30*n+30+(45*n^2+117*n+72)*x+(27*n^3+54*n^2+27*n)*x^2-(81*n^2+81*n)*x^3)*P(n,x) = -(60*n-180+(135*n^2-351*n-612)*x+(54*n^3-108*n^2-972*n-594)*x^2-(405*n^2+243*n-324)*x^3+(486*n+486)*x^4)*P(n-1,x) + (25+(75*n+135)*x+(45*n^2+315*n+162)*x^2+(162*n^2+162*n-216)*x^3+(81*n^2-405*n-567)*x^4-(243*n+243)*x^5)*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 - 180 + (135*n^2 - 351*n - 612)*x + (54*n^3 - 108*n^2 - 972*n - 594)*x^2 - (405*n^2 + 243*n - 324)*x^3 + (486*n + 486)*x^4)*P[n - 1, x] + (25 + (75*n + 135)*x + (45*n^2 + 315*n + 162)*x^2 + (162*n^2 + 162*n - 216)*x^3 + (81*n^2 - 405*n - 567)*x^4 - (243*n + 243)*x^5)*D[P[n - 1, x], x])/(30*n + 30 + (45*n^2 + 117*n + 72)*x + (27*n^3 + 54*n^2 + 27*n)*x^2 - (81*n^2 + 81*n)*x^3)]];
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 - 180 + (135*n^2 - 351*n - 612)*x + (54*n^3 - 108*n^2 - 972*n - 594)*x^2 - (405*n^2 + 243*n - 324)*x^3 + (486*n + 486)*x^4)*v[i - 1] + (25 + (75*n + 135)*x + (45*n^2 + 315*n + 162)*x^2 + (162*n^2 + 162*n - 216)*x^3 + (81*n^2 - 405*n - 567)*x^4 - (243*n + 243)*x^5)*deriv(v[i - 1], x))/(30*n + 30 + (45*n^2 + 117*n + 72)*x + (27*n^3 + 54*n^2 + 27*n)*x^2 - (81*n^2 + 81*n)*x^3)); 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.