Real-rooted atlas

A038320 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(17 + 48\,x + 24\,n x\right)\,P_{n}(x) = \left(289 + 1632\,x + 1728\,x^{2} + 408\,n x + 576\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 1632\,x + 1728\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0289163217283649
14913416168812855296189953
2835219432962996352376012816588809442177
31419857200450408489664015980544014100480047775744454947521
4241375694089188162164864320543338496071912448004873125888133772083221433397185
541033867381102231845152377081616164320256028525271040028995099033615918877900836691771392992771786369
669757574411575700504321167872138496439669510963296985921536001314444489523210824836972544499008090931299067782758445377745814273
7118587876497301352721451225526348169984112115725295616296776919900160502775017242624552066685599744381741189562368151573707620352264180754022402052125783884097
82015993900449569221807185605424348986121602722810471464960840867939717120017094350586249216234628341379891202163200074186752012883765147729920449107281838080069743719061913691994323564703041
934271896307633106444477943707211270591782274880636456947704934402246318638958592005327739266047672328775099967607930881011296034682306560803088027541831680419915308518604800130420754645778432182601737180282884093911089596482305
1058262223722976119740612273196608229920072358407552144263574813118464057281125293444096001552655443248178790429835339889866964992412608782150381076484095748940463341568028554240979265126400133029169738694000643725075438477770752474764516668735488181058251434645737089
119904578032905937363556276031370864461930327192800627231882250033699180544140656985442790502400428921066197309390848941944302237228466176151977568092057029836818103210316847969730561577621814104398233600979981550408379138048411620835951793668096104922958183790542848122708367385150095367964912076029263686593
12168377826559400929665587643811278966491616181559905457728689815955274582269952334763625353841395712011342579306106626113536280228429915575468687365167237315129939014451271809400923496946597888750947983513693559193605830890224929855871385632655252985508964335616124858320238710745989122920459143766572269568315535801847528816640348763869572760138254017
1328624230515098157931212320351227686688801797086638290453209280146585890495848732364807760429496839050536960028923577230571896589516879398055142746382794752016470568941976680608563202615913890784531626065920319152892993319762657280029737540147142264944066562081772377826196476395520106129572202904134090752037235854083023796436992080461629471119848243200807771652729673770598415209620105161139079666561
1448661191875666868481219834090355953852902434914826115357376637440306702786268237347717120175903068595018478837760071520118242868689748623362159627099882701612017254449777719469085079172546560889410722866740752862412801240136955631185363468288001348101820003782677464350721132484173537450883159162887216810909797481118171136033760507701941575436206080109427816080722993610752002197138895424712656027648205981771446066811502592660916783871403612127655425
Technical details
Node id
n-f2b1db8caed04b08 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-65bc1c81d6788de0

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:            289,             1632,              1728;
  1:           4913,            41616,             88128,             55296;
  2:          83521,           943296,           2996352,           3760128,            1658880;
  3:        1419857,         20045040,          84896640,         159805440,          141004800,           47775744;
  4:       24137569,        408918816,        2164864320,        5433384960,         7191244800,         4873125888,         1337720832;
  5:      410338673,       8110223184,       51523770816,      161643202560,       285252710400,       289950990336,       159188779008,         36691771392;
  6:     6975757441,     157570050432,     1167872138496,     4396695109632,      9698592153600,     13144444895232,     10824836972544,       4990080909312,       990677827584;
  7:   118587876497,    3013527214512,    25526348169984,   112115725295616,    296776919900160,    502775017242624,    552066685599744,     381741189562368,    151573707620352,     26418075402240;
  8:  2015993900449,   56922180718560,   542434898612160,  2722810471464960,   8408679397171200,  17094350586249216,  23462834137989120,   21632000741867520,  12883765147729920,   4491072818380800,    697437190619136;
  9: 34271896307633, 1064444779437072, 11270591782274880, 63645694770493440, 224631863895859200, 532773926604767232, 877509996760793088, 1011296034682306560, 803088027541831680, 419915308518604800, 130420754645778432, 18260173718028288;
%F Formula
The row polynomials satisfy (17+(24*n+48)*x)*P(n,x) = (289+(408*n+1632)*x+(576*n+1728)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 1632*x + 1728*x^2.
%t Mathematica
P[0, x_] := 289 + 1632*x + 1728*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((289 + (408*n + 1632)*x + (576*n + 1728)*x^2)*P[n - 1, x])/(17 + (24*n + 48)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 289 + 1632*x + 1728*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((289 + (408*n + 1632)*x + (576*n + 1728)*x^2)*v[i - 1])/(17 + (24*n + 48)*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

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.