Real-rooted atlas

A290316 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(169 - 156\,n + 36\,n^{2}\right)\,P_{n}(x) = \left(182 + 10560\,x + 1679184\,x^{2} - 240\,n - 27000\,n x - 3493854\,n x^{2} + 72\,n^{2} + 15552\,n^{2} x + 2644812\,n^{2} x^{2} - 2592\,n^{3} x - 868968\,n^{3} x^{2} + 104976\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-402\,x - 8094\,x^{2} + 1210716\,x^{3} + 720\,n x + 13284\,n x^{2} - 2340468\,n x^{3} - 216\,n^{2} x - 10368\,n^{2} x^{2} + 1605744\,n^{2} x^{3} + 2592\,n^{3} x^{2} - 462672\,n^{3} x^{3} + 46656\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-711\,x^{2} - 45018\,x^{3} + 141453\,x^{4} + 1080\,n x^{2} + 76788\,n x^{3} - 240084\,n x^{4} - 324\,n^{2} x^{2} - 42768\,n^{2} x^{3} + 131220\,n^{2} x^{4} + 7776\,n^{3} x^{3} - 23328\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-13 - 20280\,x + 84\,n + 30888\,n x - 36\,n^{2} - 15552\,n^{2} x + 2592\,n^{3} x\right)\,P_{n-1}(x) + \left(744\,x + 14196\,x^{2} - 828\,n x - 25272\,n x^{2} + 216\,n^{2} x + 14256\,n^{2} x^{2} - 2592\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
212425
311928101003
41936199263456055423
513984246888183340817820003866281
6116248234673244668800169265700107775360324072841
716537619607670772101504701630910016348962240748140624031638452131
8126196015269877011067390720196169566500104965523395216775867556005862979584003520929865903

All 15 rows

Technical details
Node id
n-f94ce942a2de57b8 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-60bf375b3a8b0a4e

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: 1,      24;
   3: 1,     192,        810;
   4: 1,     936,      19926,         34560;
   5: 1,    3984,     246888,       1833408,        1782000;
   6: 1,   16248,    2346732,      44668800,      169265700,        107775360;
   7: 1,   65376,   19607670,     772101504,     7016309100,      16348962240,       7481406240;
   8: 1,  261960,  152698770,   11067390720,   196169566500,    1049655233952,    1677586755600,      586297958400;
   9: 1, 1048368, 1141388820,  141620857920,  4365948154500,   43912964616192,  156170822310216,   183806985369600,    51200301398400;
  10: 1, 4194072, 8322316488, 1684054488960, 83975084640000, 1419321482901792, 9243140150291544, 23625662500048896, 21518879814640800, 4930399393920000;
%F Formula
The row polynomials satisfy (36*n^2-156*n+169)*P(n,x) = (72*n^2-240*n+182-(2592*n^3-15552*n^2+27000*n-10560)*x+(104976*n^4-868968*n^3+2644812*n^2-3493854*n+1679184)*x^2)*P(n-2,x) - ((216*n^2-720*n+402)*x-(2592*n^3-10368*n^2+13284*n-8094)*x^2-(46656*n^4-462672*n^3+1605744*n^2-2340468*n+1210716)*x^3)*d/dx P(n-2,x) - ((324*n^2-1080*n+711)*x^2-(7776*n^3-42768*n^2+76788*n-45018)*x^3+(23328*n^3-131220*n^2+240084*n-141453)*x^4)*d^2/dx^2 P(n-2,x) - (36*n^2-84*n+13-(2592*n^3-15552*n^2+30888*n-20280)*x)*P(n-1,x) + ((216*n^2-828*n+744)*x-(2592*n^3-14256*n^2+25272*n-14196)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 24*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 24*x;
P[n_, x_] := P[n, x] = Expand[Together[((72*n^2 - 240*n + 182 - (2592*n^3 - 15552*n^2 + 27000*n - 10560)*x + (104976*n^4 - 868968*n^3 + 2644812*n^2 - 3493854*n + 1679184)*x^2)*P[n - 2, x] - ((216*n^2 - 720*n + 402)*x - (2592*n^3 - 10368*n^2 + 13284*n - 8094)*x^2 - (46656*n^4 - 462672*n^3 + 1605744*n^2 - 2340468*n + 1210716)*x^3)*D[P[n - 2, x], x] - ((324*n^2 - 1080*n + 711)*x^2 - (7776*n^3 - 42768*n^2 + 76788*n - 45018)*x^3 + (23328*n^3 - 131220*n^2 + 240084*n - 141453)*x^4)*D[P[n - 2, x], {x, 2}] - (36*n^2 - 84*n + 13 - (2592*n^3 - 15552*n^2 + 30888*n - 20280)*x)*P[n - 1, x] + ((216*n^2 - 828*n + 744)*x - (2592*n^3 - 14256*n^2 + 25272*n - 14196)*x^2)*D[P[n - 1, x], x])/(36*n^2 - 156*n + 169)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 24*x; for(n = 3, N, my(i = n); v[i] = ((72*n^2 - 240*n + 182 - (2592*n^3 - 15552*n^2 + 27000*n - 10560)*x + (104976*n^4 - 868968*n^3 + 2644812*n^2 - 3493854*n + 1679184)*x^2)*v[i - 2] - ((216*n^2 - 720*n + 402)*x - (2592*n^3 - 10368*n^2 + 13284*n - 8094)*x^2 - (46656*n^4 - 462672*n^3 + 1605744*n^2 - 2340468*n + 1210716)*x^3)*deriv(v[i - 2], x) - ((324*n^2 - 1080*n + 711)*x^2 - (7776*n^3 - 42768*n^2 + 76788*n - 45018)*x^3 + (23328*n^3 - 131220*n^2 + 240084*n - 141453)*x^4)*deriv(deriv(v[i - 2], x), x) - (36*n^2 - 84*n + 13 - (2592*n^3 - 15552*n^2 + 30888*n - 20280)*x)*v[i - 1] + ((216*n^2 - 828*n + 744)*x - (2592*n^3 - 14256*n^2 + 25272*n - 14196)*x^2)*deriv(v[i - 1], x))/(36*n^2 - 156*n + 169)); 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. [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.