Real-rooted atlas

A062145 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-8 - 18\,x - 2\,x^{2} + 4\,n + 37\,n x + 9\,n x^{2} + 4\,n^{2} - 15\,n^{2} x - 15\,n^{2} x^{2} - 9\,n^{3} x + 10\,n^{3} x^{2} + 5\,n^{4} x - 3\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(16 - 28\,x + 20\,x^{2} + 16\,n - 52\,n x + 12\,n x^{2} - 8\,n x^{3} + 4\,n^{2} + 21\,n^{2} x + 13\,n^{2} x^{2} + 12\,n^{2} x^{3} + 6\,n^{3} x - 52\,n^{3} x^{2} + 5\,n^{4} x + 18\,n^{4} x^{2} - n^{4} x^{3} - 6\,n^{5} x^{3} + n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(4\,x - 12\,x^{2} + 8\,x^{3} + 8\,n x - 8\,n x^{2} + 4\,n x^{3} - 4\,n x^{4} - 8\,n^{2} x^{2} + 3\,n^{2} x^{3} + 5\,n^{2} x^{4} + 10\,n^{3} x^{2} - 10\,n^{3} x^{3} - n^{4} x^{3} + n^{4} x^{4} + 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2448
31040959
42018016216378
5355601134448252202
65614005040448010003612012
784302417010268801350019444962491
812058804762811760010500034020343064313742
916510560116424413952577500332640754605632811532414
102201782025660812418562494800224532090552015206487481007323056
11286286005212353294720900900011675664735735021964802843101300012134380766
1236444044990990792792028314000499458964532127621141120486486050050018634144159069748
134556552017837821761760079633125183474720226606380152216064547296751001000083853025920169727001940
1456094640306633636644608204490000596292840961920960879470592455350896130130000193793601347840351521963288223980
15680133280506797272077824486686200175207032035771435704271714304301037536812388376002882679803564288020915444664822514740156395
Technical details
Node id
n-6273849bd8391cc0 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1442d01d433060a6

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:   4,     4;
   3:  10,    40,      9;
   4:  20,   180,    162,      16;
   5:  35,   560,   1134,     448,      25;
   6:  56,  1400,   5040,    4480,    1000,      36;
   7:  84,  3024,  17010,   26880,   13500,    1944,     49;
   8: 120,  5880,  47628,  117600,  105000,   34020,   3430,     64;
   9: 165, 10560, 116424,  413952,  577500,  332640,  75460,   5632,   81;
  10: 220, 17820, 256608, 1241856, 2494800, 2245320, 905520, 152064, 8748, 100;
%F Formula
The row polynomials satisfy (4*n^2+4*n-8+(5*n^4-9*n^3-15*n^2+37*n-18)*x+(n^6-3*n^5+10*n^3-15*n^2+9*n-2)*x^2)*P(n,x) = (4*n^2+16*n+16+(5*n^4+6*n^3+21*n^2-52*n-28)*x+(n^6+18*n^4-52*n^3+13*n^2+12*n+20)*x^2+(3*n^6-6*n^5-n^4+12*n^2-8*n)*x^3)*P(n-1,x) + ((8*n+4)*x+(10*n^3-8*n^2-8*n-12)*x^2+(2*n^5-n^4-10*n^3+3*n^2+4*n+8)*x^3-(2*n^5-n^4-5*n^2+4*n)*x^4)*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[((4*n^2 + 16*n + 16 + (5*n^4 + 6*n^3 + 21*n^2 - 52*n - 28)*x + (n^6 + 18*n^4 - 52*n^3 + 13*n^2 + 12*n + 20)*x^2 + (3*n^6 - 6*n^5 - n^4 + 12*n^2 - 8*n)*x^3)*P[n - 1, x] + ((8*n + 4)*x + (10*n^3 - 8*n^2 - 8*n - 12)*x^2 + (2*n^5 - n^4 - 10*n^3 + 3*n^2 + 4*n + 8)*x^3 - (2*n^5 - n^4 - 5*n^2 + 4*n)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 4*n - 8 + (5*n^4 - 9*n^3 - 15*n^2 + 37*n - 18)*x + (n^6 - 3*n^5 + 10*n^3 - 15*n^2 + 9*n - 2)*x^2)]];
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] = ((4*n^2 + 16*n + 16 + (5*n^4 + 6*n^3 + 21*n^2 - 52*n - 28)*x + (n^6 + 18*n^4 - 52*n^3 + 13*n^2 + 12*n + 20)*x^2 + (3*n^6 - 6*n^5 - n^4 + 12*n^2 - 8*n)*x^3)*v[i - 1] + ((8*n + 4)*x + (10*n^3 - 8*n^2 - 8*n - 12)*x^2 + (2*n^5 - n^4 - 10*n^3 + 3*n^2 + 4*n + 8)*x^3 - (2*n^5 - n^4 - 5*n^2 + 4*n)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 4*n - 8 + (5*n^4 - 9*n^3 - 15*n^2 + 37*n - 18)*x + (n^6 - 3*n^5 + 10*n^3 - 15*n^2 + 9*n - 2)*x^2)); 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.