Real-rooted atlas

A038284 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(28 + 12\,x + 14\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 168 + 72\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
016872240
1352830246487200
26585684672362885184192000
3115248019756801270080362880388804800000
4193616644148928035562240152409603265920279936115200000
53162405128131898888712748804978713601600300802743372819595522688000000
65059848192151795445761951655731213940398080597445632015362887682194698241343692861440000000
77969260902427323180236840984770355235129803161618819537408064524128256138265989121693052928906992641382400000000
81239662807040478155654144081969540710408196954071040526947047424022583444889606452412825601185137049601269789696060466176030720000000000
9190908072284168181774526464015779136586752018033298956288013524974217216069557010259968248417893785606083703521280977738065920931179110403990767616675840000000000
10291568692215808137453812044595229454388295270403786992780820480324599381213184019475962872791048346841231196162555155478937605475333169152078219045273606704489594882612138803214745600000000000
11442212516527308822742357992833024536069866973921287658140956770304073846359225999360506375034692567042531875173462835293007659433328642491276591964160474528874659840610108553134084754092621824169789022208319488000000000000
1266672040953348096371458513882939392955179035698987008150099562752697958416082096009217638401240618834996789248708925048569593856303825020815540224976580424049950722325191485833216039860425471426564659010769387523327864835276810970982973446881280000000000000
13100008061430022144060004836858013286401671563312473227264028655371070969610240337724016193570406402894777281659174912018609282524951838720911475062446620672034180314841748275209765804240499507202092672337249894403261307538571264034942580770406402303906424422407052774768640147456000000000000000
1414934537173549973504960077389728212582402880232169184637747205349002599914327244806877289057032706457606484301110916551802884631643650654679859202552130174850537881601093770074935944806403645900249786482688093751720708795269121826332221599907840260904603085701120258037519535308801579821548175360451377585192963145728000000000000000
Technical details
Node id
n-746d6a84958f28b6 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5184fe74b2189306

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:            168,             72;
  1:           3528,           3024,             648;
  2:          65856,          84672,           36288,            5184;
  3:        1152480,        1975680,         1270080,          362880,           38880;
  4:       19361664,       41489280,        35562240,        15240960,         3265920,         279936;
  5:      316240512,      813189888,       871274880,       497871360,       160030080,       27433728,        1959552;
  6:     5059848192,    15179544576,     19516557312,     13940398080,      5974456320,     1536288768,      219469824,      13436928;
  7:    79692609024,   273231802368,    409847703552,    351298031616,    188195374080,    64524128256,    13826598912,    1693052928,     90699264;
  8:  1239662807040,  4781556541440,   8196954071040,   8196954071040,   5269470474240,  2258344488960,   645241282560,  118513704960,  12697896960,   604661760;
  9: 19090807228416, 81817745264640, 157791365867520, 180332989562880, 135249742172160, 69557010259968, 24841789378560, 6083703521280, 977738065920, 93117911040, 3990767616;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (14*n+28+(6*n+12)*x)*P(n-1,x) for n >= 1, with P(0,x) = 168 + 72*x.
%t Mathematica
P[0, x_] := 168 + 72*x;
P[n_, x_] := P[n, x] = Expand[Together[((14*n + 28 + (6*n + 12)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 168 + 72*x; for(n = 1, N, my(i = n + 1); v[i] = ((14*n + 28 + (6*n + 12)*x)*v[i - 1])/(n + 1)); 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.