Real-rooted atlas

A038321 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(18 + 57\,x + 39\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 324 + 1404\,x + 1521\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0324140415213249
15832379088213459319185193
210497690979229568244270968231344110556001
31889568204703208870472019219356020820969090224199601692057
43401222444215891223950274406918968160112433232609744213492351874376134296447249
56122200329285337152603546914882179474970404722195769206138854499964433617138861372310066791954897493193
61101996057619101264998414485125957126276887914752169999047691202946650159980831922043399792197612649617765352009260481111429157112001
71983592903683868006162176335227200721921694759736983045507969145194881193393314792224172379034358876816006624619038568670255001979222087283611587596351461955384057
835704672266247736012324352075426120162432043579536093849601652390743558464042962159332520064775705654614945609603974771423136078032295017812980375711050085766208140406085191601362033331456891249
96426841007923215317304402216961659374643573504010785935183227776046739052460653696014177512579731621123071794392275184576475396751185445232051501314711756566803719539395849085380161180040486793699831747583732247243920635899893042801193
101156831381426176300776159170805763584249230118768642588624443974666240126195441643764979204374775310317185945611058459812190664473620537139651211234022427810709944348546072026780683650113414736017407444372573719578468574780861654046824123815576555764251211176246293903439668001
11208229648656711685865135103830712327624675634979926016605738119890071900163281081482737889459201279621778267776889088369668513721802212403280094844639723812687361301541225395511956169615666699935316347620560135778066106075012711528023249360813523478408289728449140488347831448288074856748057971967046038752496061076057
12374813367582081024113693388166564577281601181883345784463361387690965566346534912826832533649948143718435829409791497752894464116445581822367696907008288341440703005725674496546647314666115021591232789601676739943920076224855401816468272580082576673952946308335972186272365057845917015318267564121685948639005106089188188323491941317426090413821624208892275481335249
136746640616477458432219265820035517399040332553160387201388544031223046725242797035520202949803714078180730880967394064370439328150528349336745467103090721024097315236237264432415142402108496785140729368994752035532075453297476403430080461916980892867193244591044549182387581267812257336032855206132531378644080760164276030662656893220403805084734282415570504441070734461618571137961752599217832579906859702436109193
14121439531096594251776420990374468193406156868410935851081428500480691710573605378888171520487079528913787633754112025328135503516956955213824100608982694525690127654912311408755959246183728455680759058842650662572838110720146189110436423902916969472022172015082857625275740369922620329055246810259860225536236557484154225926237381472015770498943615061749158764807322017366678421526395140802115249461484877329847485122864400312427438050835136112416457054691003038858224001
Technical details
Node id
n-85b7a886c31a0704 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a285c8a8f5f97b30

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:            324,             1404,              1521;
  1:           5832,            37908,             82134,              59319;
  2:         104976,           909792,           2956824,            4270968,            2313441;
  3:        1889568,         20470320,          88704720,          192193560,          208209690,            90224199;
  4:       34012224,        442158912,        2395027440,         6918968160,        11243323260,          9744213492,          3518743761;
  5:      612220032,       9285337152,       60354691488,       217947497040,       472219576920,        613885449996,        443361713886,        137231006679;
  6:    11019960576,     191012649984,     1448512595712,      6276887914752,     16999904769120,      29466501599808,      31922043399792,      19761264961776,       5352009260481;
  7:   198359290368,    3868006162176,    33522720072192,    169475973698304,    550796914519488,    1193393314792224,    1723790343588768,    1600662461903856,     867025500197922,     208728361158759;
  8:  3570467226624,   77360123243520,   754261201624320,   4357953609384960,  16523907435584640,   42962159332520064,   77570565461494560,   96039747714231360,   78032295017812980,   37571105008576620,    8140406085191601;
  9: 64268410079232, 1531730440221696, 16593746435735040, 107859351832277760, 467390524606536960, 1417751257973162112, 3071794392275184576, 4753967511854452320, 5150131471175656680, 3719539395849085380, 1611800404867936998, 317475837322472439;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (18+57*x+39*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 1404*x + 1521*x^2.
%t Mathematica
P[0, x_] := 324 + 1404*x + 1521*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((18 + 57*x + 39*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 324 + 1404*x + 1521*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((18 + 57*x + 39*x^2)*v[i - 1])/(1 + 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. [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.