Real-rooted atlas

delannoy → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-36\,n - 36\,n x - 459\,n x^{2} + 243\,n x^{4} + 108\,n^{2} + 54\,n^{2} x + 405\,n^{2} x^{2} + 243\,n^{2} x^{3} + 162\,n^{3} x\right)\,P_{n}(x) = \left(-264 - 1656\,x - 2646\,x^{2} + 405\,x^{3} + 1944\,x^{4} + 729\,x^{5} + 156\,n + 1368\,n x + 2997\,n x^{2} + 1377\,n x^{3} + 684\,n^{2} + 1890\,n^{2} x - 81\,n^{2} x^{2} - 729\,n^{2} x^{3} - 216\,n^{3} - 486\,n^{3} x\right)\,P_{n-1}(x) + \left(-32 - 168\,x + 252\,x^{2} + 2943\,x^{3} + 5751\,x^{4} + 3645\,x^{5} + 729\,x^{6} + 48\,n + 648\,n x + 2862\,n x^{2} + 5103\,n x^{3} + 3402\,n x^{4} + 729\,n x^{5} + 144\,n^{2} + 864\,n^{2} x + 1458\,n^{2} x^{2} + 486\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1538
22033962
36619815327444
42038941251621813050
5595347772276453234924320344
61690123393432645603296468505729132838
7469241148143982256986248184126360298892187853428

All 15 rows

Technical details
Node id
n-56c5120f2658fc1d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5a8f3aa50606b336

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:     5,      3;
  2:    20,     33,       9;
  3:    66,    198,     153,      27;
  4:   203,    894,    1251,     621,      81;
  5:   595,   3477,    7227,    6453,    2349,     243;
  6:  1690,  12339,   34326,   45603,   29646,    8505,     729;
  7:  4692,  41148,  143982,  256986,  248184,  126360,   29889,    2187;
  8: 12805, 131196,  554274, 1248966, 1622106, 1223748,  511029,  102789,   6561;
  9: 34481, 404343, 2003922, 5469822, 8984682, 9112014, 5622777, 1987983, 347733, 19683;
%F Formula
The row polynomials satisfy (108*n^2-36*n+(162*n^3+54*n^2-36*n)*x+(405*n^2-459*n)*x^2+243*n^2*x^3+243*n*x^4)*P(n,x) = -(216*n^3-684*n^2-156*n+264+(486*n^3-1890*n^2-1368*n+1656)*x+(81*n^2-2997*n+2646)*x^2+(729*n^2-1377*n-405)*x^3-1944*x^4-729*x^5)*P(n-1,x) + (144*n^2+48*n-32+(864*n^2+648*n-168)*x+(1458*n^2+2862*n+252)*x^2+(486*n^2+5103*n+2943)*x^3+(3402*n+5751)*x^4+(729*n+3645)*x^5+729*x^6)*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[(-(216*n^3 - 684*n^2 - 156*n + 264 + (486*n^3 - 1890*n^2 - 1368*n + 1656)*x + (81*n^2 - 2997*n + 2646)*x^2 + (729*n^2 - 1377*n - 405)*x^3 - 1944*x^4 - 729*x^5)*P[n - 1, x] + (144*n^2 + 48*n - 32 + (864*n^2 + 648*n - 168)*x + (1458*n^2 + 2862*n + 252)*x^2 + (486*n^2 + 5103*n + 2943)*x^3 + (3402*n + 5751)*x^4 + (729*n + 3645)*x^5 + 729*x^6)*D[P[n - 1, x], x])/(108*n^2 - 36*n + (162*n^3 + 54*n^2 - 36*n)*x + (405*n^2 - 459*n)*x^2 + 243*n^2*x^3 + 243*n*x^4)]];
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] = (-(216*n^3 - 684*n^2 - 156*n + 264 + (486*n^3 - 1890*n^2 - 1368*n + 1656)*x + (81*n^2 - 2997*n + 2646)*x^2 + (729*n^2 - 1377*n - 405)*x^3 - 1944*x^4 - 729*x^5)*v[i - 1] + (144*n^2 + 48*n - 32 + (864*n^2 + 648*n - 168)*x + (1458*n^2 + 2862*n + 252)*x^2 + (486*n^2 + 5103*n + 2943)*x^3 + (3402*n + 5751)*x^4 + (729*n + 3645)*x^5 + 729*x^6)*deriv(v[i - 1], x))/(108*n^2 - 36*n + (162*n^3 + 54*n^2 - 36*n)*x + (405*n^2 - 459*n)*x^2 + 243*n^2*x^3 + 243*n*x^4)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.