Real-rooted atlas

A038337 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(144 + 660\,x + 484\,x^{2} + 396\,n x + 484\,n x^{2} + 121\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1728 + 14256\,x + 24684\,x^{2} + 11979\,x^{3} + 4752\,n x + 13068\,n x^{2} + 7986\,n x^{3} + 1452\,n^{2} x^{2} + 1331\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 1056\,x + 1089\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0144105610892289
1172819008392042129681236
22073630412894089610222083660252653993
32488324561920188179203066624021961500579783682054248
4298598465691648338722560735989760790614000417444192868064892438254633
53583180891968307256905390081545578496022137192000175326560647291745076124717894470310610932
64299816961261279641691048624128296751071232531292608000561044994048350003763648119729178624173630693611980276087153
75159780352170272751616140475020083253415192821761147592033280015148214839296126001354913286465375645696187521149098823579476910054722354584184
861917364224227030335488021071253012480915689019801602295184066560003635571561431043780040647398402586150258278401125126894592802829537229200031384283767211488613519206529
97430083706882996800428441630904504418304015108868826726404328061382656000799825743514828899793073091317768534295852318720495055833620832018674945712720004142725457271724108488056798439963975252541044
10891610044825639230841972326444502486362357762417419012276224077905104887808000164535581523050496239503375419162624245787720546779136178220100103499520896397394210560002982762329235638459162228017896965303943956658491060891525670337241
11106993205379072510000945640243263112617022980096377117365915090944135035515138867200032084438396994846725337503795055624192639048073421625753655604671232291850243495949837421184000155103641120253196846146537853959628882741525723872444676645158021047627890647233656471144
121283918464548864659078145135083528835766383217213445759610679430479872226859665433296896005989095167439038054411208757969616810803215337153762119018086415569307945041718067211746391453735178240065143529270506342656258420611982173921286950288160805285296113676386547535996885443712556229225727061494781138189817
13154070215745863688473861866022502401223413806906998784086394160191457198080371224907072667648000107803713013902684979222417515939233621606403450859596476779069440400353632872501321728035239174361205534720002345167053738228335616116289275391978264576041701728964831711776010230874789278239712015379868260121260500106937553137040665618812653172916708603092
14184884258895036416108465431885088030721677824649472455475201275975288981521694720593959851316268236800018816648089699377741824430416306033285534842887361833805817128681472096084871889400317214720966560211050208952320007504534571962330673971244655081750519653597184200168299031192216524806547759865138073415680147646735297164100800020532010202311807795213279471930572354529483571981605484870782817
Technical details
Node id
n-becf65d0cc8ae0d7 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-40b7e3a777b181a5

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:          144,           1056,            1089;
  1:         1728,          19008,           39204,            21296;
  2:        20736,         304128,          940896,          1022208,           366025;
  3:       248832,        4561920,        18817920,         30666240,         21961500,          5797836;
  4:      2985984,       65691648,       338722560,        735989760,        790614000,        417444192,         86806489;
  5:     35831808,      919683072,      5690539008,      15455784960,      22137192000,      17532656064,       7291745076,       1247178944;
  6:    429981696,    12612796416,     91048624128,     296751071232,     531292608000,     561044994048,     350003763648,     119729178624,      17363069361;
  7:   5159780352,   170272751616,   1404750200832,    5341519282176,   11475920332800,   15148214839296,   12600135491328,    6465375645696,    1875211490988,     235794769100;
  8:  61917364224,  2270303354880,  21071253012480,   91568901980160,  229518406656000,  363557156143104,  378004064739840,  258615025827840,  112512689459280,   28295372292000,   3138428376721;
  9: 743008370688, 29968004284416, 309045044183040, 1510886882672640, 4328061382656000, 7998257435148288, 9979307309131776, 8534295852318720, 4950558336208320, 1867494571272000, 414272545727172, 41084880567984;
%F Formula
The row polynomials satisfy (144+(396*n+660)*x+(121*n^2+484*n+484)*x^2)*P(n,x) = (1728+(4752*n+14256)*x+(1452*n^2+13068*n+24684)*x^2+(1331*n^2+7986*n+11979)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 1056*x + 1089*x^2.
%t Mathematica
P[0, x_] := 144 + 1056*x + 1089*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1728 + (4752*n + 14256)*x + (1452*n^2 + 13068*n + 24684)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*P[n - 1, x])/(144 + (396*n + 660)*x + (121*n^2 + 484*n + 484)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 144 + 1056*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1728 + (4752*n + 14256)*x + (1452*n^2 + 13068*n + 24684)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*v[i - 1])/(144 + (396*n + 660)*x + (121*n^2 + 484*n + 484)*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.