Real-rooted atlas

A063967 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 - 120\,x + 16\,x^{2} + 64\,x^{3} + 8\,x^{4} - 4\,x^{5} + 108\,n x - 126\,n x^{2} - 60\,n x^{3} + 26\,n x^{4} + 6\,n x^{5} + 78\,n^{2} x^{2} - 22\,n^{2} x^{3} - 29\,n^{2} x^{4} + 18\,n^{3} x^{3} + 2\,n^{3} x^{4} - 2\,n^{3} x^{5} + n^{4} x^{4}\right)\,P_{n}(x) = \left(72 + 48\,x - 88\,x^{2} - 64\,x^{3} + 16\,x^{4} + 16\,x^{5} + 216\,n x + 276\,n x^{2} - 40\,n x^{3} - 144\,n x^{4} - 44\,n x^{5} + 156\,n^{2} x^{2} + 220\,n^{2} x^{3} + 30\,n^{2} x^{4} - 46\,n^{2} x^{5} - 12\,n^{2} x^{6} + 36\,n^{3} x^{3} + 48\,n^{3} x^{4} + 8\,n^{3} x^{5} - 4\,n^{3} x^{6} + 2\,n^{4} x^{4} + 2\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(72 + 48\,x - 88\,x^{2} - 64\,x^{3} + 16\,x^{4} + 16\,x^{5} + 216\,n x + 192\,n x^{2} - 176\,n x^{3} - 188\,n x^{4} - 36\,n x^{5} + 156\,n^{2} x^{2} + 184\,n^{2} x^{3} - 18\,n^{2} x^{4} - 54\,n^{2} x^{5} - 8\,n^{2} x^{6} + 36\,n^{3} x^{3} + 44\,n^{3} x^{4} + 4\,n^{3} x^{5} - 4\,n^{3} x^{6} + 2\,n^{4} x^{4} + 2\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
22810
36403682
416160288128592
544576154814084003976
61201952698496005600115225408
73286368284405235246400195843136156608
889620224108288249856297600191232627208192939008

All 15 rows

Technical details
Node id
n-48b8e805a964eabc · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b7c8d43fd9e53efa

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    2,      8;
   3:    6,     40,      36;
   4:   16,    160,     288,     128;
   5:   44,    576,    1548,    1408,     400;
   6:  120,   1952,    6984,    9600,    5600,    1152;
   7:  328,   6368,   28440,   52352,   46400,   19584,    3136;
   8:  896,  20224,  108288,  249856,  297600,  191232,   62720,    8192;
   9: 2448,  62976,  393120, 1090560, 1632400, 1412352,  705600,  188416,  20736;
  10: 6688, 193152, 1377216, 4463616, 8048800, 8765568, 5901952, 2400256, 539136, 51200;
%F Formula
The row polynomials satisfy (36+(108*n-120)*x+(78*n^2-126*n+16)*x^2+(18*n^3-22*n^2-60*n+64)*x^3+(n^4+2*n^3-29*n^2+26*n+8)*x^4-(2*n^3-6*n+4)*x^5)*P(n,x) = (72+(216*n+48)*x+(156*n^2+276*n-88)*x^2+(36*n^3+220*n^2-40*n-64)*x^3+(2*n^4+48*n^3+30*n^2-144*n+16)*x^4+(2*n^4+8*n^3-46*n^2-44*n+16)*x^5-(4*n^3+12*n^2)*x^6)*P(n-2,x) + (72+(216*n+48)*x+(156*n^2+192*n-88)*x^2+(36*n^3+184*n^2-176*n-64)*x^3+(2*n^4+44*n^3-18*n^2-188*n+16)*x^4+(2*n^4+4*n^3-54*n^2-36*n+16)*x^5-(4*n^3+8*n^2)*x^6)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 8*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((72 + (216*n + 48)*x + (156*n^2 + 276*n - 88)*x^2 + (36*n^3 + 220*n^2 - 40*n - 64)*x^3 + (2*n^4 + 48*n^3 + 30*n^2 - 144*n + 16)*x^4 + (2*n^4 + 8*n^3 - 46*n^2 - 44*n + 16)*x^5 - (4*n^3 + 12*n^2)*x^6)*P[n - 2, x] + (72 + (216*n + 48)*x + (156*n^2 + 192*n - 88)*x^2 + (36*n^3 + 184*n^2 - 176*n - 64)*x^3 + (2*n^4 + 44*n^3 - 18*n^2 - 188*n + 16)*x^4 + (2*n^4 + 4*n^3 - 54*n^2 - 36*n + 16)*x^5 - (4*n^3 + 8*n^2)*x^6)*P[n - 1, x])/(36 + (108*n - 120)*x + (78*n^2 - 126*n + 16)*x^2 + (18*n^3 - 22*n^2 - 60*n + 64)*x^3 + (n^4 + 2*n^3 - 29*n^2 + 26*n + 8)*x^4 - (2*n^3 - 6*n + 4)*x^5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 8*x; for(n = 3, N, my(i = n); v[i] = ((72 + (216*n + 48)*x + (156*n^2 + 276*n - 88)*x^2 + (36*n^3 + 220*n^2 - 40*n - 64)*x^3 + (2*n^4 + 48*n^3 + 30*n^2 - 144*n + 16)*x^4 + (2*n^4 + 8*n^3 - 46*n^2 - 44*n + 16)*x^5 - (4*n^3 + 12*n^2)*x^6)*v[i - 2] + (72 + (216*n + 48)*x + (156*n^2 + 192*n - 88)*x^2 + (36*n^3 + 184*n^2 - 176*n - 64)*x^3 + (2*n^4 + 44*n^3 - 18*n^2 - 188*n + 16)*x^4 + (2*n^4 + 4*n^3 - 54*n^2 - 36*n + 16)*x^5 - (4*n^3 + 8*n^2)*x^6)*v[i - 1])/(36 + (108*n - 120)*x + (78*n^2 - 126*n + 16)*x^2 + (18*n^3 - 22*n^2 - 60*n + 64)*x^3 + (n^4 + 2*n^3 - 29*n^2 + 26*n + 8)*x^4 - (2*n^3 - 6*n + 4)*x^5)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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.