Real-rooted atlas

A038321 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(81 + 315\,x + 196\,x^{2} + 189\,n x + 196\,n x^{2} + 49\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1458 + 10206\,x + 14994\,x^{2} + 6174\,x^{3} + 3402\,n x + 7938\,n x^{2} + 4116\,n x^{3} + 882\,n^{2} x^{2} + 686\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 324 + 2016\,x + 1764\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0324201617644104
15832544329525643904199424
21049761306368342921631610889604008962048
31889568293932801028764801422489608643600019361664382205952
434012224634894848277766496051209625604667544000209105971236894726415695085568
56122200321333279180869997156992161310320640196036848000131736761856464873552646746464256626259918848
6110199605762742745743361679931767808464573723443270573265280006323364569088334708957900897149085286411953891353624429773979648
719835929036855540601303043887842091212812543490532966422865737950720025609626504806418074283726643278690759081984193653039928322066104678400935684395237376
835704672266241110812026060808747644705228803225468994191360685972138521600092194655417303048133427676989440472144554491904017428773593548803718988421120003499981325209635298721298120704
964268410079232219940781160038419244818351503360798303576062361601940321191818240003042423628771000323220837360087818242337115544734924801150299057174220803681798536908800069299630239150085831373844316161314769616215474176
10115683138142617643188371573243904415688076392472576191592858254966784052388672179092480009388050054493372416115950144963161456641009633915325487513662116149087407923202650894946574336000748436006582820864125957675037229056958127119142502448440717591997054976
1120822964865671168842173245678256128884281907962168934444832728831662227456136210547665640448000274600464093931143168387604770305425440768393757226976940130304290703577729069080576155077354374598656000583780085134600273921473704797935579955222420174587934556161555680955578122241768005127314599837696
123748133675820810241632520445468619571218569920067205547622410270770605071710289923432505801174139289600768881299463007200870412209550264620901384192141752601711698446909441220955026462090138419278158986604797722624003677814536347981725696123791203026588716236828249419980797540761639203160080568680448250020153575055360064000978563735289331712
1367466406164774584323148432287689480601603856829552419613736960231092338614113481523208425241511972887347200020759795085501194423500836628650793862704152576047841503077698225831936047093979592109191053312035171543972158975180800019860198496279101318758483559062042947383459840254244779827177866854405292426610876771860480675054414652649472000398254324627999293442300087625062696571895808
141214395310965942517766044989992363802755072793404936497749111603205119584116989590975283202022057962873492963328000543529180420394908542566410549051428632458795941888153092809848634322662195201695383265315930877919232014470578091402549788672000953289527821396863300403248130019736737692872867841830562414755680641351680508072954644170098606080972078357099815239680001146972454928637965107262942851462690825830482131397870068869556600832
Technical details
Node id
n-4a80547a7752f591 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3cefa42def2d35e5

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            324,             2016,              1764;
  1:           5832,            54432,             95256,             43904;
  2:         104976,          1306368,           3429216,           3161088,             960400;
  3:        1889568,         29393280,         102876480,         142248960,           86436000,           19361664;
  4:       34012224,        634894848,        2777664960,        5120962560,         4667544000,         2091059712,          368947264;
  5:      612220032,      13332791808,       69997156992,      161310320640,       196036848000,       131736761856,        46487355264,         6746464256;
  6:    11019960576,     274274574336,     1679931767808,     4645737234432,      7057326528000,      6323364569088,      3347089579008,       971490852864,       119538913536;
  7:   198359290368,    5554060130304,    38878420912128,   125434905329664,    228657379507200,    256096265048064,    180742837266432,     78690759081984,     19365303992832,     2066104678400;
  8:  3570467226624,  111081202606080,   874764470522880,  3225468994191360,   6859721385216000,   9219465541730304,   8133427676989440,   4721445544919040,   1742877359354880,   371898842112000,   34999813252096;
  9: 64268410079232, 2199407811600384, 19244818351503360, 79830357606236160, 194032119181824000, 304242362877100032, 322083736008781824, 233711554473492480, 115029905717422080, 36817985369088000, 6929963023915008, 583137384431616;
%F Formula
The row polynomials satisfy (81+(189*n+315)*x+(49*n^2+196*n+196)*x^2)*P(n,x) = (1458+(3402*n+10206)*x+(882*n^2+7938*n+14994)*x^2+(686*n^2+4116*n+6174)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 2016*x + 1764*x^2.
%t Mathematica
P[0, x_] := 324 + 2016*x + 1764*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1458 + (3402*n + 10206)*x + (882*n^2 + 7938*n + 14994)*x^2 + (686*n^2 + 4116*n + 6174)*x^3)*P[n - 1, x])/(81 + (189*n + 315)*x + (49*n^2 + 196*n + 196)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 324 + 2016*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1458 + (3402*n + 10206)*x + (882*n^2 + 7938*n + 14994)*x^2 + (686*n^2 + 4116*n + 6174)*x^3)*v[i - 1])/(81 + (189*n + 315)*x + (49*n^2 + 196*n + 196)*x^2)); 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.