Real-rooted atlas

A249139 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-5 + 7\,x + 12\,x^{2} + 5\,n + 19\,n x + 12\,n x^{2} + 13\,n^{2} + 12\,n^{2} x + 3\,n^{3}\right)\,P_{n}(x) = \left(-5 + 7\,x + 12\,x^{2} + 20\,n + 145\,n x + 198\,n x^{2} + 72\,n x^{3} + 76\,n^{2} + 144\,n^{2} x + 72\,n^{2} x^{2} + 21\,n^{3} + 18\,n^{3} x\right)\,P_{n-1}(x) + \left(-260 - 903\,x - 1171\,x^{2} - 672\,x^{3} - 144\,x^{4} - 208\,n - 556\,n x - 492\,n x^{2} - 144\,n x^{3} - 39\,n^{2} - 75\,n^{2} x - 36\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17310
213619
346459100
410612027253
5301474216271018
673913507381082935
71933428732139188110432
848011212010719378040531825
9121243505537512179553645243106534
10300049674411874667932174151458332299
117442526660437486825903889505138517291079020
1218291771838011306889013683759217484451033389221
13448819192211533646863074760154791041334351030218710824850
1410956375076690976411810017540584415018983163061801749634020127
Technical details
Node id
n-b9aa6a5f16c078ec · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7ac04ff230617c4a

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     7,     3;
  2:    13,     6;
  3:    46,    45,     9;
  4:   106,   120,    27;
  5:   301,   474,   216,    27;
  6:   739,  1350,   738,   108;
  7:  1933,  4287,  3213,   918,   81;
  8:  4801, 12120, 10719,  3780,  405;
  9: 12124, 35055, 37512, 17955, 3645, 243;
%F Formula
The row polynomials satisfy (3*n^3+13*n^2+5*n-5+(12*n^2+19*n+7)*x+(12*n+12)*x^2)*P(n,x) = (21*n^3+76*n^2+20*n-5+(18*n^3+144*n^2+145*n+7)*x+(72*n^2+198*n+12)*x^2+72*n*x^3)*P(n-1,x) - (39*n^2+208*n+260+(75*n^2+556*n+903)*x+(36*n^2+492*n+1171)*x^2+(144*n+672)*x^3+144*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[((21*n^3 + 76*n^2 + 20*n - 5 + (18*n^3 + 144*n^2 + 145*n + 7)*x + (72*n^2 + 198*n + 12)*x^2 + 72*n*x^3)*P[n - 1, x] - (39*n^2 + 208*n + 260 + (75*n^2 + 556*n + 903)*x + (36*n^2 + 492*n + 1171)*x^2 + (144*n + 672)*x^3 + 144*x^4)*D[P[n - 1, x], x])/(3*n^3 + 13*n^2 + 5*n - 5 + (12*n^2 + 19*n + 7)*x + (12*n + 12)*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] = ((21*n^3 + 76*n^2 + 20*n - 5 + (18*n^3 + 144*n^2 + 145*n + 7)*x + (72*n^2 + 198*n + 12)*x^2 + 72*n*x^3)*v[i - 1] - (39*n^2 + 208*n + 260 + (75*n^2 + 556*n + 903)*x + (36*n^2 + 492*n + 1171)*x^2 + (144*n + 672)*x^3 + 144*x^4)*deriv(v[i - 1], x))/(3*n^3 + 13*n^2 + 5*n - 5 + (12*n^2 + 19*n + 7)*x + (12*n + 12)*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

Same fingerprint ?

References

  1. [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
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.