Real-rooted atlas

A038334 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 8\,x + 4\,n x\right)\,P_{n}(x) = \left(100 + 320\,x + 192\,x^{2} + 80\,n x + 64\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 1280\,x + 768\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
040012807682448
18000384004608016384108864
21600001024000184320013107203276804665600
33200000256000006144000065536000327680006291456194835456
4640000006144000001843200000262144000019660800007549747201174405127981535232
512800000001433600000051609600000917504000009175040000052848230400164416716802147483648322163785728
625600000000327680000000137625600000029360128000003670016000000281857228800013153337344003435973836803865470566412851722911744
75120000000007372800000000353894400000008808038400000013212057600000012683575296000078920024064000309237645312006957847019520687194767360507799783342080
810240000000000163840000000000884736000000000251658240000000044040192000000005073430118400000394600120320000020615843020800006957847019520001374389534720001209462790553619905751507009536
92048000000000003604480000000000216268800000000006920601600000000013841203200000000018602577100800000017362405294080000011338713661440000051024211476480000151182848819200002660818139217920211106232532992775105589293350912
10409600000000000078643200000000000519045120000000000184549376000000000041523609600000000006378026434560000000694496211763200000054425825574912000003061452688588800000120946279055360000031929817670615040050665495807918080365917469723852830009748476036907008
11819200000000000001703936000000000000122683392000000000004798283776000000000011995709440000000000020728585912320000000025795573579776000000023584524415795200000015919553980661760000078615081385984000000276725086478663680006586514455029350400951385421282017280630503947831869441156165046550474522624
121638400000000000000367001600000000000002862612480000000000001221381324800000000000335879864320000000000064488933949440000000009028450752921600000000943380976631808000000074291251909754880000004402444557615104000000193707560535064576000061474134913607270400013319395897948241920017654110539292344320108086391056891904044351249326690334146560
13327680000000000000007864320000000000000006606028800000000000000305345331200000000000009160359936000000000000019346680184832000000000030094835843072000000000035376786623692800000000031839107961323520000000022012222788075520000000011622453632103874560000046105601185205452800000133193958979482419200002648116580893851648000324259173170675712000184467440737095516161694900051192289077231616
146553600000000000000001677721600000000000000015099494400000000000000075161927680000000000000024427626496000000000000005628125144678400000000000963034746978304000000000012578413021757440000000000127356431845294080000000001006273041740595200000000061986419371220664320000002950758475853148979200000106555167183585935360000028246576862867750912000051881467707308113920000590295810358705651712031359464925306237747264553584558454140506734592
Technical details
Node id
n-188bb96e980930bf · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d2174dc9221c8d54

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:             400,             1280,               768;
  1:            8000,            38400,             46080,             16384;
  2:          160000,          1024000,           1843200,           1310720,             327680;
  3:         3200000,         25600000,          61440000,          65536000,           32768000,            6291456;
  4:        64000000,        614400000,        1843200000,        2621440000,         1966080000,          754974720,          117440512;
  5:      1280000000,      14336000000,       51609600000,       91750400000,        91750400000,        52848230400,        16441671680,         2147483648;
  6:     25600000000,     327680000000,     1376256000000,     2936012800000,      3670016000000,      2818572288000,      1315333734400,       343597383680,       38654705664;
  7:    512000000000,    7372800000000,    35389440000000,    88080384000000,    132120576000000,    126835752960000,     78920024064000,     30923764531200,     6957847019520,      687194767360;
  8:  10240000000000,  163840000000000,   884736000000000,  2516582400000000,   4404019200000000,   5073430118400000,   3946001203200000,   2061584302080000,   695784701952000,   137438953472000,   12094627905536;
  9: 204800000000000, 3604480000000000, 21626880000000000, 69206016000000000, 138412032000000000, 186025771008000000, 173624052940800000, 113387136614400000, 51024211476480000, 15118284881920000, 2660818139217920, 211106232532992;
%F Formula
The row polynomials satisfy (5+(4*n+8)*x)*P(n,x) = (100+(80*n+320)*x+(64*n+192)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1280*x + 768*x^2.
%t Mathematica
P[0, x_] := 400 + 1280*x + 768*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((100 + (80*n + 320)*x + (64*n + 192)*x^2)*P[n - 1, x])/(5 + (4*n + 8)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 400 + 1280*x + 768*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((100 + (80*n + 320)*x + (64*n + 192)*x^2)*v[i - 1])/(5 + (4*n + 8)*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.