Real-rooted atlas

A034870 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 8\,x + 4\,x^{2} + 12\,n x - 16\,n x^{2} + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(4 + 32\,x + 96\,x^{2} + 128\,x^{3} + 64\,x^{4} + 48\,n x + 256\,n x^{2} + 448\,n x^{3} + 256\,n x^{4} + 64\,n^{2} x^{2} + 256\,n^{2} x^{3} + 256\,n^{2} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1464144212
21651234568192640018576
3643072345601638403840004423682007041228608
42561638425804818350087168000165150722247884816777216530841670357248
51024819201658880157286408601600029727129667436544010066329609555148805242880001268776963688436736
64096393216973209611534336081100800037371248641186883174426575110144420426547204613734400033495711744144955146242835349504182124122112
71638418350085367398476336332865601536003778648473615429481267246063524249610202350878721679399321600203207317913617587891077121032067219456368293445632603979776008611350921216
8655368388608283115520469762048047710208000329772957696164581133516861418032332801748974436352038386270208000650263417323528442187717017682565377556480589269513011202899102924800087960930222081241245548544394015017074688
926214437748736144388915227380416512320864256000258744936038415261159653376683415196139522378605233438726525665935360001421290040721408246029470610227233686674043043843606329419628544295708498329600017944029765304327596422757089282003859941621762480773110169617569872503635968
1010485761677721607172259840153008209920203214028800018728204894208127455838863360665891729571840273899996577792090172838379520002400400957662822451939554906603520914352581168332801305147789960806401498256391536640001363746262163128329622135492313088050764451854417920188538756372889604398046511104000484884627849216767387032586551296
11419430473819750434879832064826781204480122725335040001272416273694729814099592478725859847220232192278113842678988801068201316188160003360561340727951368725845224309391361877470633332310016334987932756606976049442460920709120006000483553517764608592723546326486220846906353513482158082903496848142499840135459832542003200044803339613267558493660798500470784930626641749606432949299861847539712
121677721632212254721666984181764346506903552713098788864008214898047713287081546372612096475681715524730882558647352646696961123137383877836800407698870347654758412350427086714830848314049633211950039046724121413878074572812129883745880637440018401482897454478131223370242683730028134424659340132802048819221369736802328798822414954765512637153280082438144888412307456344671738481732485121027411812491565465619455550390240542721759218604441600001394663573912662573056
1367108864139586437127851737088002233382993920040131100672000050854130771558404845268570734592036162937437552640021740140904841216001073587205177344000044167377620995932160152910049645040762880448642333159928627200112068690231301242880023892195257037619200004349441412125603921920675140344196645257216089047617146229620736009921663515366942310400925771198401347584000071446392236623999795204480732600262522306560222605892706505850880084307385024375685120022869841857740800000039577633525331918848328312412835309158458319997512062460755968
14268435456601295421443652869685248112562502893568219848638464000030391875781263360317144851902627840260373149550379008017300575288484167680949277107735756800004364776141369009766401699999963700747304960565289339781510070272016137891393307378974720396978013501548134400008431224891197324525568015466851521595873165312024479999477290761191424033336789411632926162944038882390332856598528000038581051807776959889408032261274721890160607232022438673984815789768704012747276615685603590144057632001481506816000000199471272967672870993924964083682069874475008790904152160297025536606004365858973941762413390290284672490405888
Technical details
Node id
n-877e6ffbd793e3da · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-38ec58a5b1e47e30

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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:      1;
  1:      4,       64,        144;
  2:     16,      512,       3456,        8192,         6400;
  3:     64,     3072,      34560,      163840,       384000,        442368,         200704;
  4:    256,    16384,     258048,     1835008,      7168000,      16515072,       22478848,       16777216,         5308416;
  5:   1024,    81920,    1658880,    15728640,     86016000,     297271296,      674365440,     1006632960,       955514880,       524288000,        126877696;
  6:   4096,   393216,    9732096,   115343360,    811008000,    3737124864,    11868831744,    26575110144,     42042654720,     46137344000,      33495711744,      14495514624,       2835349504;
  7:  16384,  1835008,   53673984,   763363328,   6560153600,   37786484736,   154294812672,   460635242496,   1020235087872,   1679399321600,    2032073179136,    1758789107712,    1032067219456,     368293445632,      60397977600;
  8:  65536,  8388608,  283115520,  4697620480,  47710208000,  329772957696,  1645811335168,  6141803233280,  17489744363520,  38386270208000,   65026341732352,   84421877170176,   82565377556480,   58926951301120,   28991029248000,    8796093022208,   1241245548544;
  9: 262144, 37748736, 1443889152, 27380416512, 320864256000, 2587449360384, 15261159653376, 68341519613952, 237860523343872, 652566593536000, 1421290040721408, 2460294706102272, 3368667404304384, 3606329419628544, 2957084983296000, 1794402976530432, 759642275708928, 200385994162176, 24807731101696;
%F Formula
The row polynomials satisfy (1+(12*n-8)*x+(16*n^2-16*n+4)*x^2)*P(n,x) = (4+(48*n+32)*x+(64*n^2+256*n+96)*x^2+(256*n^2+448*n+128)*x^3+(256*n^2+256*n+64)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4 + (48*n + 32)*x + (64*n^2 + 256*n + 96)*x^2 + (256*n^2 + 448*n + 128)*x^3 + (256*n^2 + 256*n + 64)*x^4)*P[n - 1, x])/(1 + (12*n - 8)*x + (16*n^2 - 16*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4 + (48*n + 32)*x + (64*n^2 + 256*n + 96)*x^2 + (256*n^2 + 448*n + 128)*x^3 + (256*n^2 + 256*n + 64)*x^4)*v[i - 1])/(1 + (12*n - 8)*x + (16*n^2 - 16*n + 4)*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.