Real-rooted atlas

A056241 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 - 100\,x + 40\,x^{2} + 4\,x^{3} - 4\,x^{4} + 130\,n x - 304\,n x^{2} + 38\,n x^{3} + 28\,n x^{4} - 22\,n^{2} x + 193\,n^{2} x^{2} - 146\,n^{2} x^{3} - 4\,n^{2} x^{4} - 14\,n^{3} x^{2} + 88\,n^{3} x^{3} - 32\,n^{3} x^{4} + n^{4} x^{2} - 8\,n^{4} x^{3} + 16\,n^{4} x^{4}\right)\,P_{n}(x) = \left(-36 - 44\,x + 40\,x^{2} + 100\,x^{3} + 104\,x^{4} + 20\,x^{5} - 4\,x^{6} - 86\,n x - 130\,n x^{2} - 108\,n x^{3} - 10\,n x^{4} + 34\,n x^{5} + 12\,n x^{6} + 22\,n^{2} x - 135\,n^{2} x^{2} - 205\,n^{2} x^{3} - 223\,n^{2} x^{4} - 66\,n^{2} x^{5} + 4\,n^{2} x^{6} + 10\,n^{3} x^{2} - 46\,n^{3} x^{3} - 78\,n^{3} x^{4} - 88\,n^{3} x^{5} - 32\,n^{3} x^{6} - n^{4} x^{2} + 7\,n^{4} x^{3} - 9\,n^{4} x^{4} - 8\,n^{4} x^{5} - 16\,n^{4} x^{6}\right)\,P_{n-2}(x) + \left(72 + 88\,x - 472\,x^{2} - 472\,x^{3} + 88\,x^{4} + 72\,x^{5} + 216\,n x + 44\,n x^{2} - 516\,n x^{3} - 168\,n x^{4} + 16\,n x^{5} - 44\,n^{2} x + 290\,n^{2} x^{2} + 378\,n^{2} x^{3} - 172\,n^{2} x^{4} - 104\,n^{2} x^{5} - 24\,n^{3} x^{2} + 120\,n^{3} x^{3} + 144\,n^{3} x^{4} + 2\,n^{4} x^{2} - 14\,n^{4} x^{3} + 16\,n^{4} x^{4} + 32\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
2112922
3124541695
414017116025397
5160405720375361597
618481022562250756496206
71112144957128925579613726423431
8114423941254427675282241303423048186401
911803726248647267510465276146264963645100312485
1012205535456001691253223083314851824004981555001211112110
11126479207867235887586644811793329187203982778800079861443904639
1213121098912918470785020980083615661372979222934348074001477411123216913551773
131364148592036321315600467438498805071290515210517364526240015399672377441537919646567549
14142019656310128232732597297202462705739438528407100332702700011264253279115236909620580225158635174
Technical details
Node id
n-6fc152d0d54e7bb8 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b0345e9b1377090a

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,   4;
  2: 1,  12,    9;
  3: 1,  24,   54,    16;
  4: 1,  40,  171,   160,    25;
  5: 1,  60,  405,   720,   375,     36;
  6: 1,  84,  810,  2256,  2250,    756,    49;
  7: 1, 112, 1449,  5712,  8925,   5796,  1372,    64;
  8: 1, 144, 2394, 12544, 27675,  28224, 13034,  2304,   81;
  9: 1, 180, 3726, 24864, 72675, 104652, 76146, 26496, 3645, 100;
%F Formula
The row polynomials satisfy (36-(22*n^2-130*n+100)*x+(n^4-14*n^3+193*n^2-304*n+40)*x^2-(8*n^4-88*n^3+146*n^2-38*n-4)*x^3+(16*n^4-32*n^3-4*n^2+28*n-4)*x^4)*P(n,x) = -(36-(22*n^2-86*n-44)*x+(n^4-10*n^3+135*n^2+130*n-40)*x^2-(7*n^4-46*n^3-205*n^2-108*n+100)*x^3+(9*n^4+78*n^3+223*n^2+10*n-104)*x^4+(8*n^4+88*n^3+66*n^2-34*n-20)*x^5+(16*n^4+32*n^3-4*n^2-12*n+4)*x^6)*P(n-2,x) + (72-(44*n^2-216*n-88)*x+(2*n^4-24*n^3+290*n^2+44*n-472)*x^2-(14*n^4-120*n^3-378*n^2+516*n+472)*x^3+(16*n^4+144*n^3-172*n^2-168*n+88)*x^4+(32*n^4-104*n^2+16*n+72)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(36 - (22*n^2 - 86*n - 44)*x + (n^4 - 10*n^3 + 135*n^2 + 130*n - 40)*x^2 - (7*n^4 - 46*n^3 - 205*n^2 - 108*n + 100)*x^3 + (9*n^4 + 78*n^3 + 223*n^2 + 10*n - 104)*x^4 + (8*n^4 + 88*n^3 + 66*n^2 - 34*n - 20)*x^5 + (16*n^4 + 32*n^3 - 4*n^2 - 12*n + 4)*x^6)*P[n - 2, x] + (72 - (44*n^2 - 216*n - 88)*x + (2*n^4 - 24*n^3 + 290*n^2 + 44*n - 472)*x^2 - (14*n^4 - 120*n^3 - 378*n^2 + 516*n + 472)*x^3 + (16*n^4 + 144*n^3 - 172*n^2 - 168*n + 88)*x^4 + (32*n^4 - 104*n^2 + 16*n + 72)*x^5)*P[n - 1, x])/(36 - (22*n^2 - 130*n + 100)*x + (n^4 - 14*n^3 + 193*n^2 - 304*n + 40)*x^2 - (8*n^4 - 88*n^3 + 146*n^2 - 38*n - 4)*x^3 + (16*n^4 - 32*n^3 - 4*n^2 + 28*n - 4)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(36 - (22*n^2 - 86*n - 44)*x + (n^4 - 10*n^3 + 135*n^2 + 130*n - 40)*x^2 - (7*n^4 - 46*n^3 - 205*n^2 - 108*n + 100)*x^3 + (9*n^4 + 78*n^3 + 223*n^2 + 10*n - 104)*x^4 + (8*n^4 + 88*n^3 + 66*n^2 - 34*n - 20)*x^5 + (16*n^4 + 32*n^3 - 4*n^2 - 12*n + 4)*x^6)*v[i - 2] + (72 - (44*n^2 - 216*n - 88)*x + (2*n^4 - 24*n^3 + 290*n^2 + 44*n - 472)*x^2 - (14*n^4 - 120*n^3 - 378*n^2 + 516*n + 472)*x^3 + (16*n^4 + 144*n^3 - 172*n^2 - 168*n + 88)*x^4 + (32*n^4 - 104*n^2 + 16*n + 72)*x^5)*v[i - 1])/(36 - (22*n^2 - 130*n + 100)*x + (n^4 - 14*n^3 + 193*n^2 - 304*n + 40)*x^2 - (8*n^4 - 88*n^3 + 146*n^2 - 38*n - 4)*x^3 + (16*n^4 - 32*n^3 - 4*n^2 + 28*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 - 1, v[i])))};
T(9)

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.