Real-rooted atlas

A038305 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(169 + 780\,x + 576\,x^{2} + 468\,n x + 576\,n x^{2} + 144\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2197 + 18252\,x + 31824\,x^{2} + 15552\,x^{3} + 6084\,n x + 16848\,n x^{2} + 10368\,n x^{3} + 1872\,n^{2} x^{2} + 1728\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 169 + 1248\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0169124812962713
12197243365054427648104725
228561421824131414414376965184003720625
337129368546402847312046725120336960008957952125078125
44826809106932384555225840121485312013141440006987202561463132164041015625
562748517162180782410105110288276379084803986236800031791771648133145026562293235712126689453125
68157307212409543052817515524499257486849638410364215680001102114750464692354138112238496514048348285173763879150390625
710604499373352395671472292759480915211209935679488242522646912003223685645107227001811386368139520460718084074936532992515978035200116534423828125

All 15 rows

Technical details
Node id
n-ce8c9c3ad179c5b9 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3588fe420a4e2f99

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:           169,           1248,            1296;
  1:          2197,          24336,           50544,            27648;
  2:         28561,         421824,         1314144,          1437696,            518400;
  3:        371293,        6854640,        28473120,         46725120,          33696000,           8957952;
  4:       4826809,      106932384,       555225840,       1214853120,        1314144000,         698720256,         146313216;
  5:      62748517,     1621807824,     10105110288,      27637908480,       39862368000,       31791771648,       13314502656,        2293235712;
  6:     815730721,    24095430528,    175155244992,     574868496384,     1036421568000,     1102114750464,      692354138112,      238496514048,       34828517376;
  7:   10604499373,   352395671472,   2927594809152,   11209935679488,    24252264691200,    32236856451072,    27001811386368,    13952046071808,     4074936532992,     515978035200;
  8:  137858491849,  5090159699040,  47573415648720,  208184519761920,   525465734976000,   838158267727872,   877558870056960,   604588663111680,   264870874644480,   67077144576000,    7492001071104;
  9: 1792160394037, 72789283696272, 755888715307440, 3721298290744320, 10734514300224000, 19976105380847616, 25098183683629056, 21614044706242560, 12625511691386880, 4796015837184000, 1071356153167872, 106993205379072;
%F Formula
The row polynomials satisfy (169+(468*n+780)*x+(144*n^2+576*n+576)*x^2)*P(n,x) = (2197+(6084*n+18252)*x+(1872*n^2+16848*n+31824)*x^2+(1728*n^2+10368*n+15552)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 1248*x + 1296*x^2.
%t Mathematica
P[0, x_] := 169 + 1248*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2197 + (6084*n + 18252)*x + (1872*n^2 + 16848*n + 31824)*x^2 + (1728*n^2 + 10368*n + 15552)*x^3)*P[n - 1, x])/(169 + (468*n + 780)*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] = 169 + 1248*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2197 + (6084*n + 18252)*x + (1872*n^2 + 16848*n + 31824)*x^2 + (1728*n^2 + 10368*n + 15552)*x^3)*v[i - 1])/(169 + (468*n + 780)*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.