Real-rooted atlas

A038319 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(16 + 10\,x + 5\,n x\right)\,P_{n}(x) = \left(256 + 320\,x + 75\,x^{2} + 80\,n x + 25\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 320\,x + 75\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
025632075651
140967680360050015876
265536163840115200320003125379701
31048576327680030720001280000250000187508946126
416777216629145607372800040960000120000001800000109375208289151
5268435456117440512016515072001146880000448000000100800000122500006250004802902776
64294967296214748364803523215360029360128000143360000004300800000784000000800000003515625109866401001
768719476736386547056640724775731200704643072000412876800000154828800000376320000005760000000506250000195312502496308717826
8109951162777668719476736001449551462400016106127360000110100480000004954521600000150528000000030720000000040500000000312500000010742187556393883307251
91759218604441612094627905536028346784153600035433480192000027682406400000014533263360000052985856000000135168000000002376000000000275000000000189062500005859375001267670954343276
10281474976710656211106232532992054425825574912007559142440960000664377753600000039862665216000001695547392000000519045120000000114048000000000176000000000001815000000000112500000000317382812528372477543919901
11450359962737049636591746972385280102914288359833600157230162771968000153545080832000000103642929561600000503819796480000001799356416000000047443968000000009152000000000001258400000000001170000000000066015625000017089843750632601165979251126
1272057594037927936630503947831869440192106671605022720032017778600837120003439409810636800000257955735797760000014106954301440000005757940531200000001771241472000000004100096000000000070470400000000008736000000000007393750000000038281250000009155273437514056986374259870951
131152921504606846976108086391056891904003546584706554265600064035557201674240000750416685957120000006190937659146240000037618544803840000000172738215936000000006072827904000000000164003840000000000033825792000000000052416000000000000591500000000000045937500000000021972656250000488281250000311416313522064833376
14184467440737095516161844674407370955161606485183463413514240001261007895663738880000160088893004185600000014408000370376704000009630347469783040000004913442586624000000001943304929280000000005997854720000000000014432337920000000000268369920000000000037856000000000000039200000000000000281250000000000012500000000000025939941406256880354176878119912401
Technical details
Node id
n-fafa7a68b4008de4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-20bc6096ad8f5a15

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            256,             320,              75;
  1:           4096,            7680,            3600,             500;
  2:          65536,          163840,          115200,           32000,            3125;
  3:        1048576,         3276800,         3072000,         1280000,          250000,           18750;
  4:       16777216,        62914560,        73728000,        40960000,        12000000,         1800000,         109375;
  5:      268435456,      1174405120,      1651507200,      1146880000,       448000000,       100800000,       12250000,         625000;
  6:     4294967296,     21474836480,     35232153600,     29360128000,     14336000000,      4300800000,      784000000,       80000000,       3515625;
  7:    68719476736,    386547056640,    724775731200,    704643072000,    412876800000,    154828800000,    37632000000,     5760000000,     506250000,     19531250;
  8:  1099511627776,   6871947673600,  14495514624000,  16106127360000,  11010048000000,   4954521600000,  1505280000000,   307200000000,   40500000000,   3125000000,   107421875;
  9: 17592186044416, 120946279055360, 283467841536000, 354334801920000, 276824064000000, 145332633600000, 52985856000000, 13516800000000, 2376000000000, 275000000000, 18906250000, 585937500;
%F Formula
The row polynomials satisfy (16+(5*n+10)*x)*P(n,x) = (256+(80*n+320)*x+(25*n+75)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 320*x + 75*x^2.
%t Mathematica
P[0, x_] := 256 + 320*x + 75*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((256 + (80*n + 320)*x + (25*n + 75)*x^2)*P[n - 1, x])/(16 + (5*n + 10)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 256 + 320*x + 75*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((256 + (80*n + 320)*x + (25*n + 75)*x^2)*v[i - 1])/(16 + (5*n + 10)*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

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.