Real-rooted atlas

A139548 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-4 + 3\,x + 6\,n\right)\,P_{n}(x) = \left(16 + 72\,x + 108\,x^{2} + 54\,x^{3} + 48\,n + 144\,n x + 108\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1326018110
24482112345621603248500
3512039168120960190080155520583205832575000
453248565248258048065802241016064095800325225472139968010497636250000
55242887127040431308801526169603483648005329981445486745603695155201511654403149280018895682187500000
6498073682575360622854144282230784085399142401813089484827590492160301524664322327947776012194012160399076761668024448034012224128125000000
74613734490282393681584455684508614656017003918131246229402419293255863500814142097981441614043791360137193722265684737299046436315985305699043596288142851340806122200327343750000000
8419430400946234982499656663040650033233920293704040448097386076569602448564210892847550242488320719402375577608485258788864077474101985280538797163806722773220696064010056734392320235092492288029386561536011019960576414062500000000
93758096384960327843841155111321600868643713843245742408335360179019152621568539240618262528127714883272704024083377988567043637593550356480440274253853491242488004917329923232782982840320190236844759449683928019746816026377377634713653953726980096595077871104019835929036823046875000000000
103328599654495026151424012852689633280109477374197760658413453312000297076150134374410429269100462080291545477126553606586438980206592012132913910906880018303367271310950422619218076762112022814211336044544018629356002213888012149580001443840061962858007363584239194856278425606633134669905920120602448543744011901557422080035704672266241269531250000000000
112920577761289212704849920138403173629952131706245873664089069139158630404552069219142860818251220925808640058821077726606131215487464614186188803367350260811694080608586961771089100891724829166456012801153112138092589875212057783174766264320104243396412388147207377224976876699648420982947050029056018944232617251307526500471976507801601612899061840281602639289373920460823565083695718406426841007923269335937500000000000
122542620639232878234912686081450943031803904152562392314675201145978277656002566543166295003627522949969888933838848107688951812865392643238288055693475840081173087262716461056170981609340615524352304163565376590839808458100003951328886784584216604229427527680629523198026811703296570505398211798106112431567661147286929408269485688861201793024136695639277421199360550379810774880092161699702356804776755238156583520107233285676185978197770244627325525704704011568313814261763759765625000000000000
132199023255552082545835455283214874330739507200171255807973785600141466945600880640089248999504084992004469315282858409984018227391821795819520061633299465224847360017499454669590626304004209774472401518592000863386473612529631232015155188100645466931200228122674032443129856002944928596829971415040032549210807068105113600306892559038070705356802454012190837513912320016501116455631559065600921924725051479818240042087867882784948224001533200901444308828160430444103346664243200882276733567696896001203104636683223040090232847751241728020822964865671168202636718750000000000000
141891159999774727665795068854272149621542307758080187213954812582297616867491058601164800116522451745363722240641694065245594583040289166758393235374080010860436137345936261120344497705431940988928009318954879141996920832021648416285799022067712043398947742757473484800075313866236662200139776011331417564482610659328001478234927730231481466880166977451440259379232768016284178700830249058304001364769262545773254410240976438534880610798796800590913665074300673064960298703610916679461109760123967901763839302041600412328021084074200268801061138289554602721280020010036317315365601282518745830151584481281749129048716378112037481336758208102410864257812500000000000000
Technical details
Node id
n-c5c61f28a5dc4bc9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7939c1a278d41502

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         32,          60,            18;
  2:        448,        2112,          3456,          2160,            324;
  3:       5120,       39168,        120960,        190080,         155520,           58320,            5832;
  4:      53248,      565248,       2580480,       6580224,       10160640,         9580032,         5225472,          1399680,           104976;
  5:     524288,     7127040,      43130880,     152616960,      348364800,       532998144,       548674560,        369515520,        151165440,         31492800,          1889568;
  6:    4980736,    82575360,     622854144,    2822307840,     8539914240,     18130894848,     27590492160,      30152466432,      23279477760,      12194012160,       3990767616,        680244480,         34012224;
  7:   46137344,   902823936,    8158445568,   45086146560,   170039181312,    462294024192,    932558635008,    1414209798144,    1614043791360,    1371937222656,     847372990464,     363159853056,      99043596288,      14285134080,       612220032;
  8:  419430400,  9462349824,   99656663040,  650033233920,  2937040404480,   9738607656960,  24485642108928,   47550242488320,   71940237557760,   84852587888640,   77474101985280,   53879716380672,   27732206960640,   10056734392320,   2350924922880,    293865615360,    11019960576;
  9: 3758096384, 96032784384, 1155111321600, 8686437138432, 45742408335360, 179019152621568, 539240618262528, 1277148832727040, 2408337798856704, 3637593550356480, 4402742538534912, 4248800491732992, 3232782982840320, 1902368447594496, 839280197468160, 263773776347136, 53953726980096, 5950778711040, 198359290368;
%F Formula
The row polynomials satisfy (6*n-4+3*x)*P(n,x) = (48*n+16+(144*n+72)*x+(108*n+108)*x^2+54*x^3)*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[((48*n + 16 + (144*n + 72)*x + (108*n + 108)*x^2 + 54*x^3)*P[n - 1, x])/(6*n - 4 + 3*x)]];
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] = ((48*n + 16 + (144*n + 72)*x + (108*n + 108)*x^2 + 54*x^3)*v[i - 1])/(6*n - 4 + 3*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

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.