Real-rooted atlas

A038319 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4 + 10\,x + 5\,n x\right)\,P_{n}(x) = \left(64 + 320\,x + 300\,x^{2} + 80\,n x + 100\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 1280\,x + 1200\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0256128012002736
14096307205760032000124416
265536655360184320020480008000005412096
310485761310720049152000819200006400000019200000228427776
41677721625165824011796480002621440000307200000018432000004480000009432723456
5268435456469762048026424115200734003200001146880000001032192000005017600000010240000000383113691136
64294967296858993459205637144576001879048192000367001600000044040192000003211264000000131072000000023040000000015359376162816
7687194767361546188226560115964116992004509715660800010569646080000015854469120000015414067200000094371840000000331776000000005120000000000609359740010496
810995116277762748779069440023192823398400010307921510400002818572288000000507343011840000061656268800000005033164800000000265420800000000081920000000000011264000000000023968149773746176
9175921860444164837851162214404535485464576000226774273228800007086696038400000014882061680640000021703006617600000022145925120000000015571353600000000072089600000000000198246400000000002457600000000000935976560656121856
1028147497671065684442493013196808708132091985920048378511622144000017008070492160000004081936918118400000694496211763200000085040352460800000007474249728000000000461373440000000000019031654400000000004718592000000000005324800000000000036327590260465729536
11450359962737049614636698788954112016466286137573376001006273041740595200039307540692992000000106130359871078400000206364588638208000000294806555197440000000310928788684800000000239914188800000000000131952803840000000000490733568000000000001107558400000000000011468800000000000001402560876143198601216
127205759403792793625220157913274777603073706745680363520020491378304535756800088048891152302080000026414667345690624000005778208481869824000000943380976631808000000011608008110899200000000107481556582400000000007389357015040000000000366414397440000000000012404654080000000000002569011200000000000002457600000000000000053903826104746713808896
1311529215046068469764323455642275676160056745355304868249600040982756609071513600001921066716050227200000063395201629657497600000154085559516528640000000283014292989542400000000397988849516544000000000429926226329600000000000354689136721920000000000219848638464000000000000992372326400000000000003082813440000000000000058982400000000000000005242880000000000000002063356584060178007064576
1418446744073709551616737869762948382064640103762935414616227840008070450532247928832000040982756609071513600000014753792379265744896000003944590323623133184000000805018433392476160000000012735643184529408000000000157230162771968000000000001513340316680192000000000011256250289356800000000000635118288896000000000000026306674688000000000000007549747200000000000000001342177280000000000000001114112000000000000000078702315420581075412320256
Technical details
Node id
n-a9b3dce6b542bba6 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-eefdbaf8408bdfd8

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,            1280,             1200;
  1:           4096,           30720,            57600,             32000;
  2:          65536,          655360,          1843200,           2048000,            800000;
  3:        1048576,        13107200,         49152000,          81920000,          64000000,           19200000;
  4:       16777216,       251658240,       1179648000,        2621440000,        3072000000,         1843200000,          448000000;
  5:      268435456,      4697620480,      26424115200,       73400320000,      114688000000,       103219200000,        50176000000,        10240000000;
  6:     4294967296,     85899345920,     563714457600,     1879048192000,     3670016000000,      4404019200000,      3211264000000,      1310720000000,       230400000000;
  7:    68719476736,   1546188226560,   11596411699200,    45097156608000,   105696460800000,    158544691200000,    154140672000000,     94371840000000,     33177600000000,     5120000000000;
  8:  1099511627776,  27487790694400,  231928233984000,  1030792151040000,  2818572288000000,   5073430118400000,   6165626880000000,   5033164800000000,   2654208000000000,   819200000000000,   112640000000000;
  9: 17592186044416, 483785116221440, 4535485464576000, 22677427322880000, 70866960384000000, 148820616806400000, 217030066176000000, 221459251200000000, 155713536000000000, 72089600000000000, 19824640000000000, 2457600000000000;
%F Formula
The row polynomials satisfy (4+(5*n+10)*x)*P(n,x) = (64+(80*n+320)*x+(100*n+300)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 1280*x + 1200*x^2.
%t Mathematica
P[0, x_] := 256 + 1280*x + 1200*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((64 + (80*n + 320)*x + (100*n + 300)*x^2)*P[n - 1, x])/(4 + (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 + 1280*x + 1200*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((64 + (80*n + 320)*x + (100*n + 300)*x^2)*v[i - 1])/(4 + (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

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.