Real-rooted atlas

A038314 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 + 330\,x + 144\,x^{2} + 198\,n x + 144\,n x^{2} + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2662 + 13068\,x + 13464\,x^{2} + 3888\,x^{3} + 4356\,n x + 7128\,n x^{2} + 2592\,n x^{3} + 792\,n^{2} x^{2} + 432\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 484 + 2112\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0484211212963892
110648696968553627648193528
22342562044416376358424330245184008993680
3515363256221440137998080133816320570240008957952399171424
41133799041484246016455393664058879180803763584000118244966414631321617131827520
524943578883809564774414026124851222668484608019319731200091048624128225322352642293235712716607507328
654875873536957833428992411432995635279793065820168500681728000534151928217619828367032324036094853123482851737629369821556992
71207269217792237063773675521163767616225282633171172065283366269964288002644052044677121308672224133123995733904588868960464404485159780352001183876312245760

All 15 rows

Technical details
Node id
n-5b7c5438179c8b5d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-331d650033ed3318

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:             484,              2112,              1296;
  1:           10648,             69696,             85536,              27648;
  2:          234256,           2044416,           3763584,            2433024,             518400;
  3:         5153632,          56221440,         137998080,          133816320,           57024000,            8957952;
  4:       113379904,        1484246016,        4553936640,         5887918080,         3763584000,         1182449664,          146313216;
  5:      2494357888,       38095647744,      140261248512,       226684846080,       193197312000,        91048624128,        22532235264,         2293235712;
  6:     54875873536,      957833428992,     4114329956352,      7979306582016,      8500681728000,      5341519282176,      1982836703232,       403609485312,       34828517376;
  7:   1207269217792,    23706377367552,   116376761622528,    263317117206528,    336626996428800,    264405204467712,    130867222413312,     39957339045888,     6896046440448,      515978035200;
  8:  26559922791424,   579489224540160,  3200360944619520,   8275680826490880,  12342989869056000,  11633828996579328,   7197697232732160,   2930204863365120,   758565108449280,   113515167744000,    7492001071104;
  9: 584318301411328, 14023639233871872, 86054149844213760, 250339345001349120, 426714792615936000, 469231102862032896, 348368546064236544, 177277394233589760, 61190918748241920, 13735335297024000, 1813064259207168, 106993205379072;
%F Formula
The row polynomials satisfy (121+(198*n+330)*x+(36*n^2+144*n+144)*x^2)*P(n,x) = (2662+(4356*n+13068)*x+(792*n^2+7128*n+13464)*x^2+(432*n^2+2592*n+3888)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 2112*x + 1296*x^2.
%t Mathematica
P[0, x_] := 484 + 2112*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2662 + (4356*n + 13068)*x + (792*n^2 + 7128*n + 13464)*x^2 + (432*n^2 + 2592*n + 3888)*x^3)*P[n - 1, x])/(121 + (198*n + 330)*x + (36*n^2 + 144*n + 144)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 484 + 2112*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2662 + (4356*n + 13068)*x + (792*n^2 + 7128*n + 13464)*x^2 + (432*n^2 + 2592*n + 3888)*x^3)*v[i - 1])/(121 + (198*n + 330)*x + (36*n^2 + 144*n + 144)*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.