Real-rooted atlas

A038319 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16 + 100\,x + 100\,x^{2} + 60\,n x + 100\,n x^{2} + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(256 + 2880\,x + 6800\,x^{2} + 4500\,x^{3} + 960\,n x + 3600\,n x^{2} + 3000\,n x^{3} + 400\,n^{2} x^{2} + 500\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 2560\,x + 3600\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0256256036006416
1409661440172800128000366336
265536131072055296008192000400000019097856
3104857626214400147456000327680000320000000115200000937598976
4167772165033164803538944000104857600001536000000011059200000313600000044099997696
5268435456939524096079272345600293601280000573440000000619315200000351232000000819200000002008444502016
642949672961717986918401691143372800751619276800018350080000000264241152000002247884800000010485760000000207360000000089195832999936
7687194767363092376453120347892350976001803886264320005284823040000009512681472000001078984704000000754974720000000298598400000000512000000000003881847232659456

All 15 rows

Technical details
Node id
n-30f01afcd7e05a42 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-51a5affbafca912b

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:            256,            2560,              3600;
  1:           4096,           61440,            172800,            128000;
  2:          65536,         1310720,           5529600,           8192000,            4000000;
  3:        1048576,        26214400,         147456000,         327680000,          320000000,          115200000;
  4:       16777216,       503316480,        3538944000,       10485760000,        15360000000,        11059200000,          3136000000;
  5:      268435456,      9395240960,       79272345600,      293601280000,       573440000000,       619315200000,        351232000000,         81920000000;
  6:     4294967296,    171798691840,     1691143372800,     7516192768000,     18350080000000,     26424115200000,      22478848000000,      10485760000000,       2073600000000;
  7:    68719476736,   3092376453120,    34789235097600,   180388626432000,    528482304000000,    951268147200000,    1078984704000000,     754974720000000,     298598400000000,     51200000000000;
  8:  1099511627776,  54975581388800,   695784701952000,  4123168604160000,  14092861440000000,  30440580710400000,   43159388160000000,   40265318400000000,   23887872000000000,   8192000000000000,   1239040000000000;
  9: 17592186044416, 967570232442880, 13606456393728000, 90709709291520000, 354334801920000000, 892923700838400000, 1519210463232000000, 1771674009600000000, 1401421824000000000, 720896000000000000, 218071040000000000, 29491200000000000;
%F Formula
The row polynomials satisfy (16+(60*n+100)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (256+(960*n+2880)*x+(400*n^2+3600*n+6800)*x^2+(500*n^2+3000*n+4500)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 2560*x + 3600*x^2.
%t Mathematica
P[0, x_] := 256 + 2560*x + 3600*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((256 + (960*n + 2880)*x + (400*n^2 + 3600*n + 6800)*x^2 + (500*n^2 + 3000*n + 4500)*x^3)*P[n - 1, x])/(16 + (60*n + 100)*x + (25*n^2 + 100*n + 100)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 256 + 2560*x + 3600*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((256 + (960*n + 2880)*x + (400*n^2 + 3600*n + 6800)*x^2 + (500*n^2 + 3000*n + 4500)*x^3)*v[i - 1])/(16 + (60*n + 100)*x + (25*n^2 + 100*n + 100)*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.