Real-rooted atlas

A104684 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4 + 21\,x - 45\,x^{2} + 24\,x^{3} - 4\,x^{4} + 4\,n - 41\,n x + 125\,n x^{2} - 100\,n x^{3} + 20\,n x^{4} + 20\,n^{2} x - 116\,n^{2} x^{2} + 152\,n^{2} x^{3} - 40\,n^{2} x^{4} + 36\,n^{3} x^{2} - 100\,n^{3} x^{3} + 40\,n^{3} x^{4} + 24\,n^{4} x^{3} - 20\,n^{4} x^{4} + 4\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-44 + 183\,x + 105\,x^{2} - 112\,x^{3} + 4\,x^{4} + 28\,n - 383\,n x + 323\,n x^{2} + 556\,n x^{3} - 180\,n x^{4} - 32\,n x^{5} + 168\,n^{2} x - 728\,n^{2} x^{2} - 276\,n^{2} x^{3} + 596\,n^{2} x^{4} + 40\,n^{2} x^{5} + 300\,n^{3} x^{2} - 320\,n^{3} x^{3} - 516\,n^{3} x^{4} + 40\,n^{3} x^{5} + 184\,n^{4} x^{3} + 84\,n^{4} x^{4} - 72\,n^{4} x^{5} + 28\,n^{5} x^{4} + 24\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-2 + 18\,x + 48\,x^{2} - 12\,x^{3} - 40\,x^{4} - 24\,n x + 14\,n x^{2} + 182\,n x^{3} + 80\,n x^{4} - 80\,n x^{5} - 16\,n x^{6} - 60\,n^{2} x^{2} - 100\,n^{2} x^{3} + 128\,n^{2} x^{4} + 184\,n^{2} x^{5} + 16\,n^{2} x^{6} - 48\,n^{3} x^{3} - 136\,n^{3} x^{4} - 72\,n^{3} x^{5} + 16\,n^{3} x^{6} - 8\,n^{4} x^{4} - 24\,n^{4} x^{5} - 16\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
23811
3136436113
4634565401281187
532130725616307240012481
61683200404968043520140001152130075
789891280644012204864002580005529631361341105
8486398067923056508474432035420001282176197568819213686195

All 15 rows

Technical details
Node id
n-72c8c1d362d8a9d8 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-67545869359869df

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:       3,        8;
   3:      13,       64,        36;
   4:      63,      456,       540,       128;
   5:     321,     3072,      5616,      3072,       400;
   6:    1683,    20040,     49680,     43520,     14000,      1152;
   7:    8989,   128064,    401220,    486400,    258000,     55296,      3136;
   8:   48639,   806792,   3056508,   4744320,   3542000,   1282176,    197568,     8192;
   9:  265729,  5029888,  22357440,  42348544,  40936000,  21159936,   5619712,   655360,   20736;
  10: 1462563, 31109256, 158718528, 355117056, 422251200, 285368832, 109584384, 22413312, 2052864, 51200;
%F Formula
The row polynomials satisfy (4*n-4+(20*n^2-41*n+21)*x+(36*n^3-116*n^2+125*n-45)*x^2+(24*n^4-100*n^3+152*n^2-100*n+24)*x^3+(4*n^5-20*n^4+40*n^3-40*n^2+20*n-4)*x^4)*P(n,x) = (28*n-44+(168*n^2-383*n+183)*x+(300*n^3-728*n^2+323*n+105)*x^2+(184*n^4-320*n^3-276*n^2+556*n-112)*x^3+(28*n^5+84*n^4-516*n^3+596*n^2-180*n+4)*x^4+(24*n^5-72*n^4+40*n^3+40*n^2-32*n)*x^5)*P(n-1,x) - (2+(24*n-18)*x+(60*n^2-14*n-48)*x^2+(48*n^3+100*n^2-182*n+12)*x^3+(8*n^4+136*n^3-128*n^2-80*n+40)*x^4+(24*n^4+72*n^3-184*n^2+80*n)*x^5+(16*n^4-16*n^3-16*n^2+16*n)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((28*n - 44 + (168*n^2 - 383*n + 183)*x + (300*n^3 - 728*n^2 + 323*n + 105)*x^2 + (184*n^4 - 320*n^3 - 276*n^2 + 556*n - 112)*x^3 + (28*n^5 + 84*n^4 - 516*n^3 + 596*n^2 - 180*n + 4)*x^4 + (24*n^5 - 72*n^4 + 40*n^3 + 40*n^2 - 32*n)*x^5)*P[n - 1, x] - (2 + (24*n - 18)*x + (60*n^2 - 14*n - 48)*x^2 + (48*n^3 + 100*n^2 - 182*n + 12)*x^3 + (8*n^4 + 136*n^3 - 128*n^2 - 80*n + 40)*x^4 + (24*n^4 + 72*n^3 - 184*n^2 + 80*n)*x^5 + (16*n^4 - 16*n^3 - 16*n^2 + 16*n)*x^6)*D[P[n - 1, x], x])/(4*n - 4 + (20*n^2 - 41*n + 21)*x + (36*n^3 - 116*n^2 + 125*n - 45)*x^2 + (24*n^4 - 100*n^3 + 152*n^2 - 100*n + 24)*x^3 + (4*n^5 - 20*n^4 + 40*n^3 - 40*n^2 + 20*n - 4)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((28*n - 44 + (168*n^2 - 383*n + 183)*x + (300*n^3 - 728*n^2 + 323*n + 105)*x^2 + (184*n^4 - 320*n^3 - 276*n^2 + 556*n - 112)*x^3 + (28*n^5 + 84*n^4 - 516*n^3 + 596*n^2 - 180*n + 4)*x^4 + (24*n^5 - 72*n^4 + 40*n^3 + 40*n^2 - 32*n)*x^5)*v[i - 1] - (2 + (24*n - 18)*x + (60*n^2 - 14*n - 48)*x^2 + (48*n^3 + 100*n^2 - 182*n + 12)*x^3 + (8*n^4 + 136*n^3 - 128*n^2 - 80*n + 40)*x^4 + (24*n^4 + 72*n^3 - 184*n^2 + 80*n)*x^5 + (16*n^4 - 16*n^3 - 16*n^2 + 16*n)*x^6)*deriv(v[i - 1], x))/(4*n - 4 + (20*n^2 - 41*n + 21)*x + (36*n^3 - 116*n^2 + 125*n - 45)*x^2 + (24*n^4 - 100*n^3 + 152*n^2 - 100*n + 24)*x^3 + (4*n^5 - 20*n^4 + 40*n^3 - 40*n^2 + 20*n - 4)*x^4)); 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

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023