Real-rooted atlas

A038326 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(23\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 12\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11212
2552288840
31904419872518444100
4584016914112476928829442058000
51679046035040960274233609538560124416090037500
646341669612089131201261474560658160640171694080179159043781575000
712435014676389270024645077435104035321287680138213734402884460544250822656154414312500
83268632429121193761408896186849611827216247792332808477109043202653703700484615136870434398535686176572500000
984575864103483530123023449664463116080384672658602577924386903929856018310555533312477666666086471204968857646438023168243202542187500
102161383193755601014910369241760211807381406976025785246432153602017975807733760105285694316544036621111066624081885714186240106807453286406191736422409457876640625000
115468299480201668285302581575739206698408436995616093195247819069440850913132261068805327456132417126423162852749639680690569522970624013511142840729601566509314867208173092077568364128250664062500
121372046051396054887874351251490401922054178587345322240321523604975789568033550289214864998402450629820911878144127858947178011033647649297084973056012430251413471232021617828545167360225577341340876810699320537907213903078661718750000
1334186814113951700762140391840177845612861419939761625134976106817286541956756480125394205940557931520104676902350378795008637163753437088317442849427965681388748892916129315697459202154576911668346880337238125304610816319909684083425281390911669927936527158399256835937500
14846781088053265203445743384771143885727681797929145749390314752343951662665100755865644863260347621837721604213245319602746499072293095326581060626022415291930082490119618565983798727930916372481734434413893019238403619689211602822758451505459137431470084478735577167953921797485850368409619869816587373046875000
Technical details
Node id
n-d3d8e71722f9bca4 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-4b5617cf7459dd0c

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:            12;
  2:           552,            288;
  3:         19044,          19872,           5184;
  4:        584016,         914112,         476928,          82944;
  5:      16790460,       35040960,       27423360,        9538560,        1244160;
  6:     463416696,     1208913120,     1261474560,      658160640,      171694080,       17915904;
  7:   12435014676,    38927002464,    50774351040,    35321287680,    13821373440,     2884460544,     250822656;
  8:  326863242912,  1193761408896,  1868496118272,  1624779233280,   847710904320,   265370370048,   46151368704,   3439853568;
  9: 8457586410348, 35301230234496, 64463116080384, 67265860257792, 43869039298560, 18310555533312, 4776666660864, 712049688576, 46438023168;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (23*n+12*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 12.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 12;
P[n_, x_] := P[n, x] = Expand[Together[((23*n + 12*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] = 12; for(n = 2, N, my(i = n + 1); v[i] = ((23*n + 12*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.