Real-rooted atlas

A013613 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 6\,x + 6\,n\right)\,P_{n}(x) = \left(49 + 84\,x + 36\,x^{2} + 42\,n + 36\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
113619
213315636325
31225239414042165239
410633294002872811232129681289
5888373189904410002872808424077761228123
672270131981325741820529200025855206065284665618193357
7576480130353442671607728038548055566000217183684245696279936265474495
84529486527671044872848260810745723529646257605334336001737469442911334416796163827659537
935165286124459227107471182096131126869441450672675210417958208480090240013403335681965150721007769654653958307
10270369166921099171660733776813001494236419201966903041601740807210241041795820804115059200010052501760131010048060466176774128454229
1120620693177178443650154696272664780161430898860024654900916802596312014912191488793126498226463104033949238400073718346240864666316836279705610890820856071
1215620881269714846899087446423971405544167105439547202905756179480035503057320192311557441789441969599015014488403816793602715939072000530772092928565963407362176782336152333633493145
131176509412085121842873903665790290644101616702325654414432585560711704045329796399888046153974516249634716400656537619203590396390476616641221120211843247616003763656658944367876214784130606940162120125746145771
1488168999470379882679061514051174007039537216212813803484483507488387427024547437419956627263461714959843205538476941949952364522206893644817923351036631046435797862574081617712072704002634559661260823770463109127836416409629378885339448541
Technical details
Node id
n-fbf7a28cb848df83 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5350c1f96ee756b3

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:         1;
  1:        13,          6;
  2:       133,        156,         36;
  3:      1225,       2394,       1404,         216;
  4:     10633,      29400,      28728,       11232,        1296;
  5:     88837,     318990,     441000,      287280,       84240,        7776;
  6:    722701,    3198132,    5741820,     5292000,     2585520,      606528,      46656;
  7:   5764801,   30353442,   67160772,    80385480,    55566000,    21718368,    4245696,     279936;
  8:  45294865,  276710448,  728482608,  1074572352,   964625760,   533433600,  173746944,   29113344,   1679616;
  9: 351652861, 2445922710, 7471182096, 13112686944, 14506726752, 10417958208, 4800902400, 1340333568, 196515072, 10077696;
%F Formula
The row polynomials satisfy (6*n+1+6*x)*P(n,x) = (42*n+49+(36*n+84)*x+36*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((42*n + 49 + (36*n + 84)*x + 36*x^2)*P[n - 1, x])/(6*n + 1 + 6*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((42*n + 49 + (36*n + 84)*x + 36*x^2)*v[i - 1])/(6*n + 1 + 6*x)); 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.