Real-rooted atlas

A038319 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(8 + 10\,x + 5\,n x\right)\,P_{n}(x) = \left(128 + 320\,x + 150\,x^{2} + 80\,n x + 50\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 640\,x + 300\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02566403001196
140961536014400400037856
265536327680460800256000500001160016
3104857665536001228800010240000400000060000034730176
4167772161258291202949120003276800001920000005760000070000001021798336
5268435456234881024066060288009175040000716800000032256000007840000008000000029655914496
64294967296429496729601409286144002348810240002293760000001376256000005017600000010240000000900000000851371878656
768719476736773094113280289910292480056371445760006606028800000495452160000024084480000007372800000001296000000001000000000024223939490816
8109951162777613743895347200579820584960001288490188800001761607680000001585446912000009633792000000039321600000000103680000000001600000000000110000000000684117463550976
91759218604441624189255811072011338713661440002834678415360000442918502400000046506442752000003391094784000000173015040000000060825600000000014080000000000019360000000000120000000000019198725008859136
102814749767106564222124650659840217703302299648006047313952768000010630044057600000012756052869120000010851503308800000066437775360000000291962880000000009011200000000000185856000000000023040000000000013000000000000535870295100215296
1145035996273704967318349394477056041165715343933440012578413021757440002456721293312000000331657374597120000032244466974720000002303176212480000000121456558080000000046858240000000000012886016000000000023961600000000000270400000000000014000000000000014886917239222419456
1272057594037927936126100789566373888076842668642009088002561422288066969600055030556970188800000825458354552832000009028450752921600000073701638799360000000453437816832000000002099249152000000000072161689600000000001789132800000000000302848000000000000313600000000000001500000000000000411871376951820271616
13115292150460684697621617278211378380800141863388262170624000512284457613393920000120066669753139200000019811000509267968000002407586867445760000000221104916398080000000015546439434240000000008396996608000000000003463761100800000000001073479680000000000002422784000000000000037632000000000000003600000000000000001600000000000000011353755547780298571776
14184467440737095516163689348814741910323202594073385365405696000100880631653099110400002561422288066969600000046105601185205452800000616342238066114560000006289206510878720000000049748606189568000000000307090161664000000000001477871403008000000000054962159616000000000001550581760000000000000321126400000000000000460800000000000000004096000000000000000170000000000000000311970237665145022119936
Technical details
Node id
n-d370f98bedc4c8a7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3393477035eb5c26

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,             640,              300;
  1:           4096,           15360,            14400,             4000;
  2:          65536,          327680,           460800,           256000,            50000;
  3:        1048576,         6553600,         12288000,         10240000,          4000000,           600000;
  4:       16777216,       125829120,        294912000,        327680000,        192000000,         57600000,          7000000;
  5:      268435456,      2348810240,       6606028800,       9175040000,       7168000000,       3225600000,        784000000,         80000000;
  6:     4294967296,     42949672960,     140928614400,     234881024000,     229376000000,     137625600000,      50176000000,      10240000000,       900000000;
  7:    68719476736,    773094113280,    2899102924800,    5637144576000,    6606028800000,    4954521600000,    2408448000000,     737280000000,    129600000000,     10000000000;
  8:  1099511627776,  13743895347200,   57982058496000,  128849018880000,  176160768000000,  158544691200000,   96337920000000,   39321600000000,  10368000000000,   1600000000000,   110000000000;
  9: 17592186044416, 241892558110720, 1133871366144000, 2834678415360000, 4429185024000000, 4650644275200000, 3391094784000000, 1730150400000000, 608256000000000, 140800000000000, 19360000000000, 1200000000000;
%F Formula
The row polynomials satisfy (8+(5*n+10)*x)*P(n,x) = (128+(80*n+320)*x+(50*n+150)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 640*x + 300*x^2.
%t Mathematica
P[0, x_] := 256 + 640*x + 300*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((128 + (80*n + 320)*x + (50*n + 150)*x^2)*P[n - 1, x])/(8 + (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 + 640*x + 300*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((128 + (80*n + 320)*x + (50*n + 150)*x^2)*v[i - 1])/(8 + (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.