Real-rooted atlas

A063007 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-48 - 162\,x + 1053\,x^{2} + 2187\,x^{3} - 972\,x^{4} + 48\,n + 162\,n x - 2673\,n x^{2} - 5103\,n x^{3} + 3888\,n x^{4} + 2160\,n^{2} x^{2} + 3888\,n^{2} x^{3} - 6075\,n^{2} x^{4} - 540\,n^{3} x^{2} - 972\,n^{3} x^{3} + 4617\,n^{3} x^{4} - 1701\,n^{4} x^{4} + 243\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-144 - 630\,x - 81\,x^{2} + 2673\,x^{3} + 2916\,x^{4} + 144\,n + 126\,n x - 5103\,n x^{2} - 16929\,n x^{3} - 14580\,n x^{4} + 504\,n^{2} x + 6804\,n^{2} x^{2} + 19440\,n^{2} x^{3} + 12393\,n^{2} x^{4} - 5832\,n^{2} x^{5} - 1620\,n^{3} x^{2} - 4536\,n^{3} x^{3} + 3645\,n^{3} x^{4} + 11664\,n^{3} x^{5} - 648\,n^{4} x^{3} - 5103\,n^{4} x^{4} - 7290\,n^{4} x^{5} + 729\,n^{5} x^{4} + 1458\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(16 + 216\,x + 684\,x^{2} - 162\,x^{3} - 3078\,x^{4} - 2916\,x^{5} + 1008\,n x^{2} + 7128\,n x^{3} + 16200\,n x^{4} + 11664\,n x^{5} - 504\,n^{2} x^{2} - 3564\,n^{2} x^{3} - 6804\,n^{2} x^{4} + 5832\,n^{2} x^{6} - 1296\,n^{3} x^{4} - 5832\,n^{3} x^{5} - 5832\,n^{3} x^{6} + 324\,n^{4} x^{4} + 1458\,n^{4} x^{5} + 1458\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
232427
313216486715
46315847290864017577
53211080075330181440141750409641
61683709206633902419200382725022044969186939
78989455112534114026127360625117507274836833006204200198923
848639287548840599468249298560799895250140646844812872419564803701764266797985

All 15 rows

Technical details
Node id
n-2869b22ec20dd5b7 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9aff797b8fdd9c6c

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,        24;
   3:      13,       216,        486;
   4:      63,      1584,       7290,        8640;
   5:     321,     10800,      75330,      181440,      141750;
   6:    1683,     70920,     663390,     2419200,     3827250,      2204496;
   7:    8989,    455112,    5341140,    26127360,    62511750,     72748368,     33006204;
   8:   48639,   2875488,   40599468,   249298560,   799895250,   1406468448,   1287241956,    480370176;
   9:  265729,  17965152,  296484300,  2191432320,  8826914250,  20806033248,  28748403684,  21616657920,   6839645670;
  10: 1462563, 111292920, 2102100660, 18170127360, 88114776750, 260705901456, 482715733500, 547622000640, 348821929170, 95698746000;
%F Formula
The row polynomials satisfy (48*n-48+(162*n-162)*x-(540*n^3-2160*n^2+2673*n-1053)*x^2-(972*n^3-3888*n^2+5103*n-2187)*x^3+(243*n^5-1701*n^4+4617*n^3-6075*n^2+3888*n-972)*x^4)*P(n,x) = (144*n-144+(504*n^2+126*n-630)*x-(1620*n^3-6804*n^2+5103*n+81)*x^2-(648*n^4+4536*n^3-19440*n^2+16929*n-2673)*x^3+(729*n^5-5103*n^4+3645*n^3+12393*n^2-14580*n+2916)*x^4+(1458*n^5-7290*n^4+11664*n^3-5832*n^2)*x^5)*P(n-1,x) + (16+216*x-(504*n^2-1008*n-684)*x^2-(3564*n^2-7128*n+162)*x^3+(324*n^4-1296*n^3-6804*n^2+16200*n-3078)*x^4+(1458*n^4-5832*n^3+11664*n-2916)*x^5+(1458*n^4-5832*n^3+5832*n^2)*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[((144*n - 144 + (504*n^2 + 126*n - 630)*x - (1620*n^3 - 6804*n^2 + 5103*n + 81)*x^2 - (648*n^4 + 4536*n^3 - 19440*n^2 + 16929*n - 2673)*x^3 + (729*n^5 - 5103*n^4 + 3645*n^3 + 12393*n^2 - 14580*n + 2916)*x^4 + (1458*n^5 - 7290*n^4 + 11664*n^3 - 5832*n^2)*x^5)*P[n - 1, x] + (16 + 216*x - (504*n^2 - 1008*n - 684)*x^2 - (3564*n^2 - 7128*n + 162)*x^3 + (324*n^4 - 1296*n^3 - 6804*n^2 + 16200*n - 3078)*x^4 + (1458*n^4 - 5832*n^3 + 11664*n - 2916)*x^5 + (1458*n^4 - 5832*n^3 + 5832*n^2)*x^6)*D[P[n - 1, x], x])/(48*n - 48 + (162*n - 162)*x - (540*n^3 - 2160*n^2 + 2673*n - 1053)*x^2 - (972*n^3 - 3888*n^2 + 5103*n - 2187)*x^3 + (243*n^5 - 1701*n^4 + 4617*n^3 - 6075*n^2 + 3888*n - 972)*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] = ((144*n - 144 + (504*n^2 + 126*n - 630)*x - (1620*n^3 - 6804*n^2 + 5103*n + 81)*x^2 - (648*n^4 + 4536*n^3 - 19440*n^2 + 16929*n - 2673)*x^3 + (729*n^5 - 5103*n^4 + 3645*n^3 + 12393*n^2 - 14580*n + 2916)*x^4 + (1458*n^5 - 7290*n^4 + 11664*n^3 - 5832*n^2)*x^5)*v[i - 1] + (16 + 216*x - (504*n^2 - 1008*n - 684)*x^2 - (3564*n^2 - 7128*n + 162)*x^3 + (324*n^4 - 1296*n^3 - 6804*n^2 + 16200*n - 3078)*x^4 + (1458*n^4 - 5832*n^3 + 11664*n - 2916)*x^5 + (1458*n^4 - 5832*n^3 + 5832*n^2)*x^6)*deriv(v[i - 1], x))/(48*n - 48 + (162*n - 162)*x - (540*n^3 - 2160*n^2 + 2673*n - 1053)*x^2 - (972*n^3 - 3888*n^2 + 5103*n - 2187)*x^3 + (243*n^5 - 1701*n^4 + 4617*n^3 - 6075*n^2 + 3888*n - 972)*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