Real-rooted atlas

A038314 → x ↦ x+1 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(34 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1156 + 1632\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0115616325763364
139304832325875213824195112
2133633637731843995136188006433177611316496
345435424160360320226391040159805440564019207962624656356768
41544804416654270105611545943040108667699205752995840162437529619110297638068692544
5525233501442595271418885495868887046465728102404564043366401933006602244548250828845864714242207984167552
61785793904896100844832276482491460562124835173560877056310354948915201752592652697661856211271681247520227328110075314176128063081718016
76071699276646438573148345753610891241885859841793851604729856189937228736102413407333793136646309333549711361908705947811843368304613785626418075402247427658739644928
82064377754059776145720782639513604628777801490432087129935086878720107631096283791360911698697933291525362933517254656021632000741867520572611784343552089821456367616063403380965376430804206899405824
97018884363803238454499572707178086419235143308415795204073324465311580160575057571573399552056829218837841838084011474270906482688202259206936461312071385602448162816016796612340744192023712864481050624152168114316902424986644000165537792
102386420683693101056202143869677533265927847938469833644441618465737576079163392029327936150243377152033123316122627814195227278025042164082278416504351286015243059272813314497126072320228433927834121011204837424354134327296620845906412961792365203474360565761449225352009601191936
11811383032455654359047445632533122475294723153444366969518948352816185600862699021926414403275309341747445760183006321577518673428481722412438376646338150412158205447364562386944643669700154594479308825241949025670371737607127138548424575549441372069453172645560321614199356673700659287648833846535782484055070416556869132288
12275870231034922482073627262469890509986463744125086626556457584951296353185769100586122215424685595904724667178418176967900100787765428355072102483540083410457119948882675797042079024231219251064462878931162025164824030335472438193894195284812948726252449038336217701686570059762237443841794468883407568896417208449109510324224210357201231685877764875194084160298409672704
13937958785518736439050249931328317257209354649604907244580291797563473920150103951867749101941555203178671921905275099938816049362905140176036846108672580740060472659257013043205270582061432537794740224037204108668935560904048640204257851515724648100659208650920770077749801910272277569650376826196852736065310505971017928671232010638815452292513267712010728217262815979765760504857282956046106624282761256881297307761016832
14318905987076370389277081636017617363919479259529216190681503691338419609272320628127306277350088124661760144099793793039137863892992024412200360232512767530106883159225928971266358150955008318577404602144506704297984025298793894876181414753075201587375303207917265239408640784350149820382648706531328301995779609986902175776768888222881205843829928755201929171868682375739211776029180750954859464962867202746423619280890820034561211657479094510655897616400152899115243850138976256
Technical details
Node id
n-6520f6faa376ffcc · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-8b732aeffa021ba5

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:              1156,               1632,                 576;
  1:             39304,              83232,               58752,               13824;
  2:           1336336,            3773184,             3995136,             1880064,              331776;
  3:          45435424,          160360320,           226391040,           159805440,            56401920,             7962624;
  4:        1544804416,         6542701056,         11545943040,         10866769920,          5752995840,          1624375296,           191102976;
  5:       52523350144,       259527141888,        549586888704,        646572810240,        456404336640,        193300660224,         45482508288,          4586471424;
  6:     1785793904896,     10084483227648,      24914605621248,      35173560877056,      31035494891520,      17525926526976,       6185621127168,       1247520227328,       110075314176;
  7:    60716992766464,    385731483457536,    1089124188585984,    1793851604729856,    1899372287361024,    1340733379313664,     630933354971136,     190870594781184,     33683046137856,      2641807540224;
  8:  2064377754059776,  14572078263951360,   46287778014904320,   87129935086878720,  107631096283791360,   91169869793329152,   53629335172546560,   21632000741867520,   5726117843435520,    898214563676160,    63403380965376;
  9: 70188843638032384, 544995727071780864, 1923514330841579520, 4073324465311580160, 5750575715733995520, 5682921883784183808, 4011474270906482688, 2022592069364613120, 713856024481628160, 167966123407441920, 23712864481050624, 1521681143169024;
%F Formula
The row polynomials satisfy P(n,x) = (34+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 1156 + 1632*x + 576*x^2.
%t Mathematica
P[0, x_] := 1156 + 1632*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(34 + 24*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1156 + 1632*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (34 + 24*x)*v[i - 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.