Real-rooted atlas

A038320 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(289 + 1020\,x + 576\,x^{2} + 612\,n x + 576\,n x^{2} + 144\,n^{2} x^{2}\right)\,P_{n}(x) = \left(4913 + 31212\,x + 41616\,x^{2} + 15552\,x^{3} + 10404\,n x + 22032\,n x^{2} + 10368\,n x^{3} + 2448\,n^{2} x^{2} + 1728\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 1632\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0289163212963217
14913416166609627648140273
283521943296224726418800645184005672545
31419857200450406367248079902720440640008957952218062049
4241375694089188161623648240271669248022472640009137111041463132168080685425
5410338673811022318438642828112808216012808914147200054365810688174112727042293235712291196782353
6697575744115757005043287590410387221983475548163030810048000246458341785611839665438723118800568323482851737610264866050497
711858787649730135272145121914476112748856057862647808927427874688009427031573299260382293737472238588243476485328763158528515978035200355433701346945

All 15 rows

Technical details
Node id
n-854249d2fdc28473 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-95a0f773c9801a91

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:            289,             1632,             1296;
  1:           4913,            41616,            66096,             27648;
  2:          83521,           943296,          2247264,           1880064,            518400;
  3:        1419857,         20045040,         63672480,          79902720,          44064000,           8957952;
  4:       24137569,        408918816,       1623648240,        2716692480,        2247264000,         913711104,         146313216;
  5:      410338673,       8110223184,      38642828112,       80821601280,       89141472000,       54365810688,       17411272704,        2293235712;
  6:     6975757441,     157570050432,     875904103872,     2198347554816,     3030810048000,     2464583417856,     1183966543872,      311880056832,       34828517376;
  7:   118587876497,    3013527214512,   19144761127488,    56057862647808,    92742787468800,    94270315732992,    60382293737472,    23858824347648,     5328763158528,     515978035200;
  8:  2015993900449,   56922180718560,  406826173959120,  1361405235732480,  2627712311616000,  3205190734921728,  2566247483842560,  1352000046366720,   452944868474880,   87716265984000,    7492001071104;
  9: 34271896307633, 1064444779437072, 8452943836706160, 31822847385246720, 70197457467456000, 99895111238393856, 95977655895711744, 63206002167644160, 28233563468267520, 8201470869504000, 1401004200296448, 106993205379072;
%F Formula
The row polynomials satisfy (289+(612*n+1020)*x+(144*n^2+576*n+576)*x^2)*P(n,x) = (4913+(10404*n+31212)*x+(2448*n^2+22032*n+41616)*x^2+(1728*n^2+10368*n+15552)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 1632*x + 1296*x^2.
%t Mathematica
P[0, x_] := 289 + 1632*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4913 + (10404*n + 31212)*x + (2448*n^2 + 22032*n + 41616)*x^2 + (1728*n^2 + 10368*n + 15552)*x^3)*P[n - 1, x])/(289 + (612*n + 1020)*x + (144*n^2 + 576*n + 576)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 289 + 1632*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4913 + (10404*n + 31212)*x + (2448*n^2 + 22032*n + 41616)*x^2 + (1728*n^2 + 10368*n + 15552)*x^3)*v[i - 1])/(289 + (612*n + 1020)*x + (144*n^2 + 576*n + 576)*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. [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.