Real-rooted atlas

A178822 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 6\,n x + 18\,n^{2} x + 9\,n^{3} x^{2}\right)\,P_{n}(x) = \left(40 + 180\,x + 270\,x^{2} + 135\,x^{3} + 8\,n + 216\,n x + 504\,n x^{2} + 297\,n x^{3} + 36\,n^{2} x + 180\,n^{2} x^{2} + 189\,n^{2} x^{3} + 18\,n^{3} x^{2} + 27\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1127284
284100817012793
34488064272162419259920
4201648384244944435456255150985950
58064241920163296043545605103000220449613545000
6295681064448898128031933440561330004849891216503102163143750
71013764257792431101441916006404490640005819869443960744481108546561777050000
832947215814656186810624996323328291891600050438868485148967824288222105668396456717877234375
9102502455351296747242496464950886416345929600353072079364805703302440351094784191510078763940536600168605937500
10307507218450432028021593601992646656081729648000211843247616360427747680403510947840287265118140118216098000214562217871507365234375
1189456645904138249963233280797058662403736212480001129830653952230673758515232280875827203064161260160189145756800068659909718411142404582412882187500000
12253460481824915456338749931523011110502401587890304000548774889062413071512982528219509955624962604537071136021436519104000116721846521283788417558016555766117452105933242187500
1370189056547474636811086361395210839997808646351561216000246948700078086722492391014413170597337497618752666912179219292867193600014006621582553668191516044288200075802282722677391254944842575781250000
1419051315216003104768351068110848374472651571224135932620800104267228921856319318388573184714975284035584118766890443801614662579067136001330629050342592863759203227648380144024337168101740867687872125136817947006509522460937500
Technical details
Node id
n-e16a97073df8bd83 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-56805ed74e931bf6

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:      12,       72;
  2:      84,     1008,      1701;
  3:     448,     8064,     27216,      24192;
  4:    2016,    48384,    244944,     435456,      255150;
  5:    8064,   241920,   1632960,    4354560,     5103000,     2204496;
  6:   29568,  1064448,   8981280,   31933440,    56133000,    48498912,    16503102;
  7:  101376,  4257792,  43110144,  191600640,   449064000,   581986944,   396074448,   110854656;
  8:  329472, 15814656, 186810624,  996323328,  2918916000,  5043886848,  5148967824,  2882221056,   683964567;
  9: 1025024, 55351296, 747242496, 4649508864, 16345929600, 35307207936, 48057033024, 40351094784, 19151007876, 3940536600;
%F Formula
The row polynomials satisfy (4*n+(18*n^2-6*n)*x+9*n^3*x^2)*P(n,x) = (8*n+40+(36*n^2+216*n+180)*x+(18*n^3+180*n^2+504*n+270)*x^2+(27*n^3+189*n^2+297*n+135)*x^3)*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[((8*n + 40 + (36*n^2 + 216*n + 180)*x + (18*n^3 + 180*n^2 + 504*n + 270)*x^2 + (27*n^3 + 189*n^2 + 297*n + 135)*x^3)*P[n - 1, x])/(4*n + (18*n^2 - 6*n)*x + 9*n^3*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] = ((8*n + 40 + (36*n^2 + 216*n + 180)*x + (18*n^3 + 180*n^2 + 504*n + 270)*x^2 + (27*n^3 + 189*n^2 + 297*n + 135)*x^3)*v[i - 1])/(4*n + (18*n^2 - 6*n)*x + 9*n^3*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

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.