Real-rooted atlas

delannoy → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(32 - 16\,x - 16\,x^{2} - 2\,x^{3} + 2\,x^{4} + 96\,n x - 8\,n x^{2} - 28\,n x^{3} - 4\,n x^{4} + 88\,n^{2} x^{2} + 20\,n^{2} x^{3} - 8\,n^{2} x^{4} - n^{2} x^{5} + 24\,n^{3} x^{3} + 4\,n^{3} x^{4} - n^{3} x^{5} + 2\,n^{4} x^{4}\right)\,P_{n}(x) = \left(32 + 112\,x + 144\,x^{2} + 78\,x^{3} + 10\,x^{4} - 6\,x^{5} - 2\,x^{6} + 96\,n x + 264\,n x^{2} + 252\,n x^{3} + 84\,n x^{4} - 5\,n x^{5} - 5\,n x^{6} + 88\,n^{2} x^{2} + 180\,n^{2} x^{3} + 108\,n^{2} x^{4} + 12\,n^{2} x^{5} - 4\,n^{2} x^{6} + 24\,n^{3} x^{3} + 36\,n^{3} x^{4} + 11\,n^{3} x^{5} - n^{3} x^{6} + 2\,n^{4} x^{4} + 2\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(64 + 96\,x + 16\,x^{2} - 60\,x^{3} - 36\,x^{4} - 7\,x^{5} - x^{6} + 192\,n x + 256\,n x^{2} + 96\,n x^{3} - 20\,n x^{4} - 16\,n x^{5} - 3\,n x^{6} + 176\,n^{2} x^{2} + 200\,n^{2} x^{3} + 60\,n^{2} x^{4} - 5\,n^{2} x^{5} - 3\,n^{2} x^{6} + 48\,n^{3} x^{3} + 40\,n^{3} x^{4} + 6\,n^{3} x^{5} - n^{3} x^{6} + 4\,n^{4} x^{4} + 2\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
2520934
312727216172
42923236017625822
57070014761136350363768
616920285409571227756124916754
7408571218432248001660057609806472756

All 15 rows

Technical details
Node id
n-af0a275e6fe209dd · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-26b1aa13355c20b9

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,     4;
  2:    5,    20,      9;
  3:   12,    72,     72,     16;
  4:   29,   232,    360,    176,     25;
  5:   70,   700,   1476,   1136,    350,     36;
  6:  169,  2028,   5409,   5712,   2775,    612,    49;
  7:  408,  5712,  18432,  24800,  16600,   5760,   980,    64;
  8:  985, 15760,  59688,  97696,  83650,  40032, 10682,  1472,   81;
  9: 2378, 42804, 186120, 359072, 375300, 230184, 84672, 18240, 2106, 100;
%F Formula
The row polynomials satisfy (32+(96*n-16)*x+(88*n^2-8*n-16)*x^2+(24*n^3+20*n^2-28*n-2)*x^3+(2*n^4+4*n^3-8*n^2-4*n+2)*x^4-(n^3+n^2)*x^5)*P(n,x) = (32+(96*n+112)*x+(88*n^2+264*n+144)*x^2+(24*n^3+180*n^2+252*n+78)*x^3+(2*n^4+36*n^3+108*n^2+84*n+10)*x^4+(2*n^4+11*n^3+12*n^2-5*n-6)*x^5-(n^3+4*n^2+5*n+2)*x^6)*P(n-2,x) + (64+(192*n+96)*x+(176*n^2+256*n+16)*x^2+(48*n^3+200*n^2+96*n-60)*x^3+(4*n^4+40*n^3+60*n^2-20*n-36)*x^4+(2*n^4+6*n^3-5*n^2-16*n-7)*x^5-(n^3+3*n^2+3*n+1)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((32 + (96*n + 112)*x + (88*n^2 + 264*n + 144)*x^2 + (24*n^3 + 180*n^2 + 252*n + 78)*x^3 + (2*n^4 + 36*n^3 + 108*n^2 + 84*n + 10)*x^4 + (2*n^4 + 11*n^3 + 12*n^2 - 5*n - 6)*x^5 - (n^3 + 4*n^2 + 5*n + 2)*x^6)*P[n - 2, x] + (64 + (192*n + 96)*x + (176*n^2 + 256*n + 16)*x^2 + (48*n^3 + 200*n^2 + 96*n - 60)*x^3 + (4*n^4 + 40*n^3 + 60*n^2 - 20*n - 36)*x^4 + (2*n^4 + 6*n^3 - 5*n^2 - 16*n - 7)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*P[n - 1, x])/(32 + (96*n - 16)*x + (88*n^2 - 8*n - 16)*x^2 + (24*n^3 + 20*n^2 - 28*n - 2)*x^3 + (2*n^4 + 4*n^3 - 8*n^2 - 4*n + 2)*x^4 - (n^3 + n^2)*x^5)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((32 + (96*n + 112)*x + (88*n^2 + 264*n + 144)*x^2 + (24*n^3 + 180*n^2 + 252*n + 78)*x^3 + (2*n^4 + 36*n^3 + 108*n^2 + 84*n + 10)*x^4 + (2*n^4 + 11*n^3 + 12*n^2 - 5*n - 6)*x^5 - (n^3 + 4*n^2 + 5*n + 2)*x^6)*v[i - 2] + (64 + (192*n + 96)*x + (176*n^2 + 256*n + 16)*x^2 + (48*n^3 + 200*n^2 + 96*n - 60)*x^3 + (4*n^4 + 40*n^3 + 60*n^2 - 20*n - 36)*x^4 + (2*n^4 + 6*n^3 - 5*n^2 - 16*n - 7)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*v[i - 1])/(32 + (96*n - 16)*x + (88*n^2 - 8*n - 16)*x^2 + (24*n^3 + 20*n^2 - 28*n - 2)*x^3 + (2*n^4 + 4*n^3 - 8*n^2 - 4*n + 2)*x^4 - (n^3 + n^2)*x^5)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.