Real-rooted atlas

A136124 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 12\,x + 81\,x^{2} - 2\,n - 12\,n x - n^{2}\right)\,P_{n-2}(x) + \left(12\,x^{2} + 36\,x^{3} + 12\,n x^{2}\right)\,P_{n-2}'(x) + \left(3 + 12\,x + 2\,n\right)\,P_{n-1}(x) + \left(-12\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131215
2128481177
3605649724322028
4360410496397776202523904
525203304895175105840506258748295956
6201602946249844741354320901125288684357213879108
718144028935361084897817439408144585006491016150028213996853953128
8181440031112640128021148232255296226332225127370880415078027278336531441796224168
919958400364011840161824294832375877123578351175237883489297668358224270451233480783196830012451824144
1023950080046076428802187599529647481681600581164065004400449578021433778793673945920013232880901476225007144929205977016368
11311351040062773367040315489528288733933835712978083667000823121321220458324148744171598388544427916293206820159500628753752255091683596703818688
124358914560091618926336048405736235521195768451750417133486192900157490199385209777616810731419825981836812506204419832539697490003355973151326019351368981352966157260981696
1365383718400014265908697600787928818809602051816604215043130539390693003102519594311282109727502432551012861917175683469957437761184414737979001425306161565158845589136105081828933124873081279195149241728
141046139494400023610058536960013569809938041603703135270109184597065205833460063164243676774244642426171569186244724780312812893976419921941726358052290570053447448471417763022580465672739977137141560811964107616802525948621078088832
Technical details
Node id
n-e3fc2db31bb04e25 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-11954419b7b7488a

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:       12,        84,         81;
  3:       60,       564,        972,        432;
  4:      360,      4104,       9639,       7776,       2025;
  5:     2520,     33048,      95175,     105840,      50625,       8748;
  6:    20160,    294624,     984474,    1354320,     901125,     288684,     35721;
  7:   181440,   2893536,   10848978,   17439408,   14458500,    6491016,   1500282,    139968;
  8:  1814400,  31112640,  128021148,  232255296,  226332225,  127370880,  41507802,   7278336,   531441;
  9: 19958400, 364011840, 1618242948, 3237587712, 3578351175, 2378834892, 976683582, 242704512, 33480783, 1968300;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n+1+(12*n+12)*x-81*x^2)*P(n-2,x) + ((12*n+12)*x^2+36*x^3)*d/dx P(n-2,x) + (2*n+3+12*x)*P(n-1,x) - 12*x^2*d/dx 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[-(n^2 + 2*n + 1 + (12*n + 12)*x - 81*x^2)*P[n - 2, x] + ((12*n + 12)*x^2 + 36*x^3)*D[P[n - 2, x], x] + (2*n + 3 + 12*x)*P[n - 1, x] - 12*x^2*D[P[n - 1, x], x]];
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] = -(n^2 + 2*n + 1 + (12*n + 12)*x - 81*x^2)*v[i - 2] + ((12*n + 12)*x^2 + 36*x^3)*deriv(v[i - 2], x) + (2*n + 3 + 12*x)*v[i - 1] - 12*x^2*deriv(v[i - 1], x)); 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.