Real-rooted atlas

A038277 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(18\,n + 22\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 22\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
12222
27929681760
3213845227231944105600
4513216188179222999689370245632000
511547360564537601034985608433216025768160281600000
6249422976152425152037259481604553936640278296128068027942413516800000
75237882496384111383041173673670401912653388801753265606408571520742417460505216630784000000
81077507256329218673192963380180170752688555219968084156749107206171494934528251431275110443900698828828835840000000
92181952194048213346436751369126486461030422309189126963234083483388416033326072646451220365933283942471119132102656108654229601281297612800000000
1043639043880960480029482690560234681080426496066927567380889601227005401982976014996732690903040122195599703654406400721889239040195577613282304026559922791424057671680000000000
1186405306884300810560648619192320580835674055577601893094048773734404049117826543820805938706145597603846048682185330892804224476446897766401936218371494809605258864712701952064275013155246082537553920000000000
121696686026091724822811001017455411213940056177333862405111353931689082880124944207219066470402137934212415137382426130307040629456896228121728132479385601394077227476262912056795738897181081601388340284153315328154260031572590592110729625600000000000
13330853775087886336485252203462233292832619731454961238016132895202223916154880365461806115769425920714680865293060210688101908197458454881894410676096876600035246088155351780736138035204430067633980124364801624358132459378933763609684738798619852836765307524800757764798283776000000000000
146413473178626719744101902962727068991488747288393331839270912334895909604268710297610232930571241543925760225124472567313966366723668695108504375748198444839606881720148033536411029729749101356974082790942609407478349824013644608312658783043584454820277088626101452892648574962497909555287105497827989487616206695301120000000000000
Technical details
Node id
n-1f17a7a2f7f54d5c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6bae7ffb6b8d3725

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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:             0;
  1:            22;
  2:           792,            968;
  3:         21384,          52272,          31944;
  4:        513216,        1881792,        2299968,          937024;
  5:      11547360,       56453760,      103498560,        84332160,        25768160;
  6:     249422976,     1524251520,     3725948160,      4553936640,      2782961280,       680279424;
  7:    5237882496,    38411138304,   117367367040,    191265338880,    175326560640,     85715207424,     17460505216;
  8:  107750725632,   921867319296,  3380180170752,   6885552199680,   8415674910720,   6171494934528,   2514312751104,   439006988288;
  9: 2181952194048, 21334643675136, 91264864610304, 223091891269632, 340834833884160, 333260726464512, 203659332839424, 71119132102656, 10865422960128;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (18*n+22*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 22.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 22;
P[n_, x_] := P[n, x] = Expand[Together[((18*n + 22*n*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] = 0; v[2] = 22; for(n = 2, N, my(i = n + 1); v[i] = ((18*n + 22*n*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.