Real-rooted atlas

A038323 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(20\,n + 18\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 18\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11818
27206481368
321600388801749677976
4576000155520013996804199043950784
5144000005184000069984000419904009447840187662240
634560000015552000002799360000251942400011337408002040733448557398144
78064000000435456000009797760000011757312000079361856000285702681604285540224379377984384
8184320000000116121600000031352832000004702924800000423263232000022856214528006856864358408815968460816475843893248
94147200000000298598400000009405849600000016930529280000019046845440000013713728716800061711779225600158687432294401785233613312704342326436352
10921600000000007464960000000002687385600000000564350976000000076187381760000006856864358400000411411861504000015868743229440003570467226624003570467226624029738898227312640
11202752000000000018247680000000000739031040000000001773674496000000002793537331200000003017020317696000002262765238272000001163707836825600003927513949286400078550278985728007069525108715521243085945901668352
12442368000000000004379443200000000001970749440000000000532102348800000000095778422784000000001206808127078400000010861273143705600000698224702095360000031420111594291200009426033478287360001696686026091724801388197657711411251531562848287342592
139584640000000000001035141120000000000051239485440000000000153718456320000000000311279874048000000000448243018629120000000470655169560576000000363076845089587200000204230725362892800000816922901451571200002205691833919242240036093139100496691202706985432537251842121382670587828936704
1420643840000000000000241532928000000000000130427781120000000000043041167769600000000009684262748160000000000156885056520192000000001882620678242304000000016943586104180736000000114369206203219968000005718460310160998400000205864571165795942400050530394740695367680075795592111043051520524738714614913433686813506211748076486656
Technical details
Node id
n-a76100adbc274ca7 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-249d425e0def4c4f

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:            18;
  2:           720,            648;
  3:         21600,          38880,          17496;
  4:        576000,        1555200,        1399680,          419904;
  5:      14400000,       51840000,       69984000,        41990400,         9447840;
  6:     345600000,     1555200000,     2799360000,      2519424000,      1133740800,       204073344;
  7:    8064000000,    43545600000,    97977600000,    117573120000,     79361856000,     28570268160,     4285540224;
  8:  184320000000,  1161216000000,  3135283200000,   4702924800000,   4232632320000,   2285621452800,   685686435840,    88159684608;
  9: 4147200000000, 29859840000000, 94058496000000, 169305292800000, 190468454400000, 137137287168000, 61711779225600, 15868743229440, 1785233613312;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (20*n+18*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 18.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 18;
P[n_, x_] := P[n, x] = Expand[Together[((20*n + 18*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] = 18; for(n = 2, N, my(i = n + 1); v[i] = ((20*n + 18*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.