Real-rooted atlas

A134346 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 5\,x + 5\,x^{2} - x^{3} + 6\,n x - 17\,n x^{2} + 5\,n x^{3} + 11\,n^{2} x^{2} - 11\,n^{2} x^{3} + n^{2} x^{4} + 6\,n^{3} x^{3} - 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(-2 - 6\,x - 4\,x^{2} + 4\,x^{3} + 6\,x^{4} + 2\,x^{5} - 12\,n x - 34\,n x^{2} - 34\,n x^{3} - 14\,n x^{4} - 2\,n x^{5} - 22\,n^{2} x^{2} - 58\,n^{2} x^{3} - 52\,n^{2} x^{4} - 18\,n^{2} x^{5} - 2\,n^{2} x^{6} - 12\,n^{3} x^{3} - 28\,n^{3} x^{4} - 20\,n^{3} x^{5} - 4\,n^{3} x^{6} - 2\,n^{4} x^{4} - 4\,n^{4} x^{5} - 2\,n^{4} x^{6}\right)\,P_{n-2}(x) + \left(3 - 3\,x - 24\,x^{2} - 24\,x^{3} - 3\,x^{4} + 3\,x^{5} + 18\,n x - 36\,n x^{3} - 18\,n x^{4} + 33\,n^{2} x^{2} + 27\,n^{2} x^{3} - 12\,n^{2} x^{4} - 6\,n^{2} x^{5} + 18\,n^{3} x^{3} + 18\,n^{3} x^{4} + 3\,n^{4} x^{4} + 3\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131215
275663126
315180405240840
431496167419847754960
56312605670100807875226827216
61273048171454064047625274326223142240
72557140481951428002231251927808746516320718080
8511163521287724578568942501030176701092261632413913532032
91023368283314521374912322245046403284210668235699274576710230017022720
102047818808290353930240107467501857038421063630157209607461315204700024768780700928
1140951801802027025108108003378375068108040927026108648640054729675225225005450445589680377395200
128191393168486545428832320101363625233541792370855716415185408328418145180202000654133261415404813842791744617472
1316383851916115008667496860829284612575905715613775481721799246592170787860111713845005669500981840138563599345132110687985467392
1432767183495226836173190835008819994175236158322448215657497197206016797034338165599534003968771807171751507250392369389912648737257536237680640
Technical details
Node id
n-d7ab911fc2275448 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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:    3,    12;
  2:    7,    56,     63;
  3:   15,   180,    405,     240;
  4:   31,   496,   1674,    1984,     775;
  5:   63,  1260,   5670,   10080,    7875,    2268;
  6:  127,  3048,  17145,   40640,   47625,   27432,    6223;
  7:  255,  7140,  48195,  142800,  223125,  192780,   87465,   16320;
  8:  511, 16352, 128772,  457856,  894250, 1030176,  701092,  261632,  41391;
  9: 1023, 36828, 331452, 1374912, 3222450, 4640328, 4210668, 2356992, 745767, 102300;
%F Formula
The row polynomials satisfy (1+(6*n-5)*x+(11*n^2-17*n+5)*x^2+(6*n^3-11*n^2+5*n-1)*x^3+(n^4-2*n^3+n^2)*x^4)*P(n,x) = -(2+(12*n+6)*x+(22*n^2+34*n+4)*x^2+(12*n^3+58*n^2+34*n-4)*x^3+(2*n^4+28*n^3+52*n^2+14*n-6)*x^4+(4*n^4+20*n^3+18*n^2+2*n-2)*x^5+(2*n^4+4*n^3+2*n^2)*x^6)*P(n-2,x) + (3+(18*n-3)*x+(33*n^2-24)*x^2+(18*n^3+27*n^2-36*n-24)*x^3+(3*n^4+18*n^3-12*n^2-18*n-3)*x^4+(3*n^4-6*n^2+3)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2 + (12*n + 6)*x + (22*n^2 + 34*n + 4)*x^2 + (12*n^3 + 58*n^2 + 34*n - 4)*x^3 + (2*n^4 + 28*n^3 + 52*n^2 + 14*n - 6)*x^4 + (4*n^4 + 20*n^3 + 18*n^2 + 2*n - 2)*x^5 + (2*n^4 + 4*n^3 + 2*n^2)*x^6)*P[n - 2, x] + (3 + (18*n - 3)*x + (33*n^2 - 24)*x^2 + (18*n^3 + 27*n^2 - 36*n - 24)*x^3 + (3*n^4 + 18*n^3 - 12*n^2 - 18*n - 3)*x^4 + (3*n^4 - 6*n^2 + 3)*x^5)*P[n - 1, x])/(1 + (6*n - 5)*x + (11*n^2 - 17*n + 5)*x^2 + (6*n^3 - 11*n^2 + 5*n - 1)*x^3 + (n^4 - 2*n^3 + n^2)*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] = 3 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2 + (12*n + 6)*x + (22*n^2 + 34*n + 4)*x^2 + (12*n^3 + 58*n^2 + 34*n - 4)*x^3 + (2*n^4 + 28*n^3 + 52*n^2 + 14*n - 6)*x^4 + (4*n^4 + 20*n^3 + 18*n^2 + 2*n - 2)*x^5 + (2*n^4 + 4*n^3 + 2*n^2)*x^6)*v[i - 2] + (3 + (18*n - 3)*x + (33*n^2 - 24)*x^2 + (18*n^3 + 27*n^2 - 36*n - 24)*x^3 + (3*n^4 + 18*n^3 - 12*n^2 - 18*n - 3)*x^4 + (3*n^4 - 6*n^2 + 3)*x^5)*v[i - 1])/(1 + (6*n - 5)*x + (11*n^2 - 17*n + 5)*x^2 + (6*n^3 - 11*n^2 + 5*n - 1)*x^3 + (n^4 - 2*n^3 + n^2)*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

Same fingerprint ?

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.