Real-rooted atlas

A038279 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 4\,x + 4\,n\right)\,P_{n}(x) = \left(81 + 72\,x + 16\,x^{2} + 36\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
113417
215310416273
317011836624644225
4182252721614688332825663713
519026936450027216097920166401024942513
6194861745664564374000217728058752079872409613737841
71966331754561276639303844082400015240960329011237273616384197899169
819610172962922614487298041668192409632659200097542144175472641703936655362823683265
91937102445705966224411326070592104757649926137316864235146240058525286490243072766771226214439970805329
10189836039617748409780014119324488015101427456010475764992049098534912156764160003344302080451215360340787201048576562038466769
111847995732538352785742841704650151600207083425824016611570201609218673192963600559226889853747200018393661440220594176014994636841943047857934035393
12178872039771388703795161442004668578281627274402425600248500110988801594710739353673749385543682468954898432591224832000980995276801058852044865431142416777216109321784036257
13172282017253419301346068107623062986741974434747588690214435456723153280025844011542835213820826407731254785257832448160482068398083415965696000510117543936500548239362835349504671088641514375532961265
14165221278841385964779296619096260437689907012843050908585018884864662416630016397115299316736024121077439979521105666112618496383496804827136998555092254721912940789760025969620418562335891783681221381324826843545620898382354865457
Technical details
Node id
n-6888d2debd0d217d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-835fdf194c1ce669

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,          4;
  2:        153,        104,          16;
  3:       1701,       1836,         624,          64;
  4:      18225,      27216,       14688,        3328,        256;
  5:     190269,     364500,      272160,       97920,      16640,       1024;
  6:    1948617,    4566456,     4374000,     2177280,     587520,      79872,      4096;
  7:   19663317,   54561276,    63930384,    40824000,   15240960,    3290112,    372736,    16384;
  8:  196101729,  629226144,   872980416,   681924096,  326592000,   97542144,  17547264,  1703936,   65536;
  9: 1937102445, 7059662244, 11326070592, 10475764992, 6137316864, 2351462400, 585252864, 90243072, 7667712, 262144;
%F Formula
The row polynomials satisfy (4*n+5+4*x)*P(n,x) = (36*n+81+(16*n+72)*x+16*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[((36*n + 81 + (16*n + 72)*x + 16*x^2)*P[n - 1, x])/(4*n + 5 + 4*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] = ((36*n + 81 + (16*n + 72)*x + 16*x^2)*v[i - 1])/(4*n + 5 + 4*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

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.