Real-rooted atlas

A038301 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(40 + 88\,x + 20\,n + 44\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1760 + 3872\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0176038725632
152800232320255552540672
214080009292800204441601499238446137344
335200000309760000102220800014992384008245811203690987520
4844800000929280000040888320000899543040009894973440043537883136283467841536
5197120000002601984000001431091200000419786752000069264814080006095303639040223494466764821165598834688
64505600000006938624000000457949184000001679147008000003694123417600004876242911232003575911468236801123857890017281548112371908608
71013760000000017842176000000013738475520000006044929228800000166235553792000002925745746739200032183203214131200202294420203110405563096555585536111464090777419776
822528000000000044605440000000003925278720000000020149764096000000066494221516800000014628728733696000002145546880942080000202294420203110400011126193111171072002719736093841817607926335344172072960
949561600000000001090355200000000001079451648000000000633278300160000000024381214556160000000643664064282624000001180050784518144000001483492414822809600001223881242228817920005983419406451998720013163522694194397184558014008229713936384
1010813440000000000026168524800000000002878537728000000000018998349004800000000083592735621120000000025746562571304960000005664243765687091200000890095448893685760000097910499378305433600007180103287742398464000315924544660665532416063184908932133106483238959523483674573012992
1123429120000000000006185287680000000000074841980928000000000054884119347200000000002716763907686400000000095630089550561280000000245450563179773952000000462849633424716595200000636418245958985318400000622275618271007866880000410701908058865192140800164280763223546076856320301181399243167807569922701193628201437062234112
125046272000000000000014432337920000000000001905068605440000000000015367553417216000000000084521543794688000000000033470531342696448000000009818022527190958080000000215996495598201077760000003563942177370317783040000043559293278970550681600000383321780854940845998080002299930685129645075988480084330791788086986119577601427136476413779765100544186174576220652892904751104
13108134400000000000000033305395200000000000000476267151360000000000000419115093196800000000000025356463138406400000000000111568437808988160000000000368175844769660928000000000925699266849433190400000000178197108868515889152000000026135575967382330408960000002874913356412056344985600000229993068512964507598848000012649618768213047917936640004281409429241339295301632006727929103093533178331136012766256655130484084897218560
142306867200000000000000076126617600000000000000011723499110400000000000000111764024852480000000000000737642564026368000000000000357019000988762112000000000013090696702921277440000000000370279706739773276160000000008146153548275012075520000000013938973849270576218112000000018399445481037160607907840000018399445481037160607907840000013492926686093917779132416000068502550867861428724826112000215293731298993061706596352003157641392385231571696746496871509787656907713528983453696
Technical details
Node id
n-78a31084f0075a3e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2235b7dd30e0ad6a

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:             1760,               3872;
  1:            52800,             232320,              255552;
  2:          1408000,            9292800,            20444160,            14992384;
  3:         35200000,          309760000,          1022208000,          1499238400,            824581120;
  4:        844800000,         9292800000,         40888320000,         89954304000,          98949734400,          43537883136;
  5:      19712000000,       260198400000,       1431091200000,       4197867520000,        6926481408000,        6095303639040,         2234944667648;
  6:     450560000000,      6938624000000,      45794918400000,     167914700800000,      369412341760000,      487624291123200,       357591146823680,       112385789001728;
  7:   10137600000000,    178421760000000,    1373847552000000,    6044929228800000,    16623555379200000,    29257457467392000,     32183203214131200,     20229442020311040,      5563096555585536;
  8:  225280000000000,   4460544000000000,   39252787200000000,  201497640960000000,   664942215168000000,  1462872873369600000,   2145546880942080000,   2022944202031104000,   1112619311117107200,   271973609384181760;
  9: 4956160000000000, 109035520000000000, 1079451648000000000, 6332783001600000000, 24381214556160000000, 64366406428262400000, 118005078451814400000, 148349241482280960000, 122388124222881792000, 59834194064519987200, 13163522694194397184;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (20*n+40+(44*n+88)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1760 + 3872*x.
%t Mathematica
P[0, x_] := 1760 + 3872*x;
P[n_, x_] := P[n, x] = Expand[Together[((20*n + 40 + (44*n + 88)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1760 + 3872*x; for(n = 1, N, my(i = n + 1); v[i] = ((20*n + 40 + (44*n + 88)*x)*v[i - 1])/(n + 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.