Real-rooted atlas

A062264 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36\,n + 9\,n^{2} - 24\,n^{2} x + 26\,n^{3} x + 8\,n^{4} x + 4\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(144 - 144\,x + 24\,x^{2} + 72\,n + 120\,n x - 92\,n x^{2} + 9\,n^{2} + 188\,n^{2} x - 25\,n^{2} x^{2} - 12\,n^{2} x^{3} + 58\,n^{3} x + 94\,n^{3} x^{2} - 4\,n^{3} x^{3} + 8\,n^{4} x + 55\,n^{4} x^{2} + 19\,n^{4} x^{3} + 8\,n^{5} x^{2} + 14\,n^{5} x^{3} + n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(36\,x - 48\,x^{2} + 12\,x^{3} + 18\,n x + 16\,n x^{2} - 34\,n x^{3} + 44\,n^{2} x^{2} - 32\,n^{2} x^{3} - 12\,n^{2} x^{4} + 16\,n^{3} x^{2} + 4\,n^{3} x^{3} - 20\,n^{3} x^{4} + 10\,n^{4} x^{3} - 10\,n^{4} x^{4} + 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1549
21548972
33525218916492
4708961512512253015
512625207560576011253617127
6210604828350384001687521604992092
7330129368731818480014437541580377364475176
849525344232848709632866250443520905526144812374866
9715463325559842306304405405032432401177176179712947710011573090
101001800801216215658944015765750181621441030029027955203316951400012155256256
11136513213224774751698840053088750832431606798191428828800608107557750019965144259420680
1218202096644756752402688001592662503261772803625702082214051847297290012320000958320276481691200934995
132380321776868795288994048434541250112633092016352656321359181824645080436170170000235320801527552373491965493673395
143060479808152039161853429761095043950350414064064388584266990077952451556305217153136003706302604277145623529874939222524875831700
Technical details
Node id
n-ab6c501992bb4229 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-95a0709737b5bfee

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:   5,     4;
  2:  15,    48,      9;
  3:  35,   252,    189,      16;
  4:  70,   896,   1512,     512,      25;
  5: 126,  2520,   7560,    5760,    1125,      36;
  6: 210,  6048,  28350,   38400,   16875,    2160,      49;
  7: 330, 12936,  87318,  184800,  144375,   41580,    3773,     64;
  8: 495, 25344, 232848,  709632,  866250,  443520,   90552,   6144,   81;
  9: 715, 46332, 555984, 2306304, 4054050, 3243240, 1177176, 179712, 9477, 100;
%F Formula
The row polynomials satisfy (9*n^2+36*n+(8*n^4+26*n^3-24*n^2)*x+(n^6+4*n^5)*x^2)*P(n,x) = (9*n^2+72*n+144+(8*n^4+58*n^3+188*n^2+120*n-144)*x+(n^6+8*n^5+55*n^4+94*n^3-25*n^2-92*n+24)*x^2+(3*n^6+14*n^5+19*n^4-4*n^3-12*n^2)*x^3)*P(n-1,x) + ((18*n+36)*x+(16*n^3+44*n^2+16*n-48)*x^2+(2*n^5+10*n^4+4*n^3-32*n^2-34*n+12)*x^3-(2*n^5+10*n^4+20*n^3+12*n^2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^2 + 72*n + 144 + (8*n^4 + 58*n^3 + 188*n^2 + 120*n - 144)*x + (n^6 + 8*n^5 + 55*n^4 + 94*n^3 - 25*n^2 - 92*n + 24)*x^2 + (3*n^6 + 14*n^5 + 19*n^4 - 4*n^3 - 12*n^2)*x^3)*P[n - 1, x] + ((18*n + 36)*x + (16*n^3 + 44*n^2 + 16*n - 48)*x^2 + (2*n^5 + 10*n^4 + 4*n^3 - 32*n^2 - 34*n + 12)*x^3 - (2*n^5 + 10*n^4 + 20*n^3 + 12*n^2)*x^4)*D[P[n - 1, x], x])/(9*n^2 + 36*n + (8*n^4 + 26*n^3 - 24*n^2)*x + (n^6 + 4*n^5)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((9*n^2 + 72*n + 144 + (8*n^4 + 58*n^3 + 188*n^2 + 120*n - 144)*x + (n^6 + 8*n^5 + 55*n^4 + 94*n^3 - 25*n^2 - 92*n + 24)*x^2 + (3*n^6 + 14*n^5 + 19*n^4 - 4*n^3 - 12*n^2)*x^3)*v[i - 1] + ((18*n + 36)*x + (16*n^3 + 44*n^2 + 16*n - 48)*x^2 + (2*n^5 + 10*n^4 + 4*n^3 - 32*n^2 - 34*n + 12)*x^3 - (2*n^5 + 10*n^4 + 20*n^3 + 12*n^2)*x^4)*deriv(v[i - 1], x))/(9*n^2 + 36*n + (8*n^4 + 26*n^3 - 24*n^2)*x + (n^6 + 4*n^5)*x^2)); 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.