Real-rooted atlas

A038337 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(22 + 23\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 484 + 1012\,x + 529\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
048410125292025
11064833396349141216791125
2234256979616153621610706962798414100625
35153632269394405632792058888280307825106436343184528125
41133799047112012161858821360259108432020316456608495972761480358898303765625
5249435788818254164544572516978889975674632010429114388065418990252227975269063404825447373669453125
654875873536458961851392167938313804835114374704644588810330720383791409478420061823677285992492786727831098528116815125390625
71207269217792113593058219524750255161907211587743652531218171688909651218997674769180813240803627004859325678588528155055750856381801152661463756680642578125
82655992279142427767192009216013063201695244803641862290795520666295260020544083589768984395527282441994852640435054976315872017056132594201803962535855218604142651121364934050628916015625
958431830141132867196604662302723512549789165824011016633429656448023034778989281664033714540157039526435247019255086777626320826067110256013758613625989452047946683848145060100252157137030589528097579139271532278301220703125
1012855002631049216161271851189526528927313144339777536323154580603255808076014770664629491201271519800208347852815508688472238182144138973961634342151689080684993153038320421930817863676528013233284742088036562515417760892767282191462443202032168952523554931640625
11282810057883082752384364578668371558424110141752834215936924222100525311610882415580490009337164804545683285744843573766336407004371600133126624425504570309230085194151816083537919043016805347725287175201261573145412392818723597047398076657210462675825875578118065040363619364673833102863559971923828125
126221821273427820544910648386383526461446188269716560782090242587821881470872510464743998790922875846758415556338355660131341312243951669668306605125122914747222010936061235226663312655895494651072185835209419877689992329714113219675424704144369296866202536806934496520771848390301812415524319947643195396411592836324538749809139628860198736572265625
13136880068015412051968214652833933259808768015708684665115831459840711651017404489940377602231996372768627540275205133591657367843342632968944894554504575521254401202333229079511125259520125698473949221617640768010220936518093272949577606411314724985780304735043046699146170928657208801061728490332293319936402561512791361127240406038256359870977874369702666352354643912456076283298708943145751953125
14301136149633906514329650371865029671635124224394961214437198048133120192693198558754199240704065471893601213074514739201642749330357709869642547231486028831856105834815488470245885151097684545945605530732853765751176193792051396709348126172546447360376130463865832444544455682144876198904333774674995293432107149241812154403203005508341863722628743040673311933729210588906720938556028834657184536646132610415680998648961282748441902441558837890625
Technical details
Node id
n-6f980c37a6fafca3 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-efa2c9a7f1c36a29

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:             484,             1012,               529;
  1:           10648,            33396,             34914,              12167;
  2:          234256,           979616,           1536216,            1070696,             279841;
  3:         5153632,         26939440,          56327920,           58888280,           30782510,            6436343;
  4:       113379904,        711201216,        1858821360,         2591084320,         2031645660,          849597276,          148035889;
  5:      2494357888,      18254164544,       57251697888,        99756746320,       104291143880,        65418990252,        22797526906,         3404825447;
  6:     54875873536,     458961851392,     1679383138048,      3511437470464,      4588810330720,      3837914094784,      2006182367728,       599249278672,        78310985281;
  7:   1207269217792,   11359305821952,    47502551619072,    115877436525312,    181716889096512,    189976747691808,    132408036270048,     59325678588528,     15505575085638,     1801152661463;
  8:  26559922791424,  277671920092160,  1306320169524480,   3641862290795520,   6662952600205440,   8358976898439552,   7282441994852640,   4350549763158720,   1705613259420180,   396253585521860,    41426511213649;
  9: 584318301411328, 6719660466230272, 35125497891658240, 110166334296564480, 230347789892816640, 337145401570395264, 352470192550867776, 263208260671102560, 137586136259894520, 47946683848145060, 10025215713703058, 952809757913927;
%F Formula
The row polynomials satisfy P(n,x) = (22+23*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1012*x + 529*x^2.
%t Mathematica
P[0, x_] := 484 + 1012*x + 529*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 23*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 484 + 1012*x + 529*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 23*x)*v[i - 1]); 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.