Real-rooted atlas

A038335 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 - 21\,x + 63\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1029 + 1323\,x + 567\,x^{2} + 81\,x^{3} + 1323\,n x + 945\,n x^{2} + 162\,n x^{3} + 189\,n^{2} x^{2} + 81\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213657
244115127292682
39261476284592711664114480
4194481133358419289349797761640254600800
54084101350065806751269051438240172226252125764177390000
685766121882165816212664973521604060801085025375267846264260406096633900000
7180108854121613062492625235022097939492344053166243375196867004043827969523306110016242319600000
83782285936151871349980817506580618522667669427584223298222175011024552226243215494399325142648268834867844018686764000000
9794280046581122546064329644726776767000484031586968896844067279821505209100926898420257614715716485980261401665900225178938742048900306661140000000
10166798809782012859408167691601240778901337605252094760906688029542354793752502187822389297328106352477257509034018618298112069195236437845813583026900042190091252110687869000000000
1135027750054222166052328673675963184665846766519572792362211806160974897708193832508423116198794712849134844492969158196457520671596805328033205714065939688396069500974591107923514518872583696368465760000000000
1273558275113866411513198802342394728025357933851629142038186141930572480307092778081057237530323218315660966082063663468704704636990145904184847872335666091959986095789338252698380001227984795983622611387558910913924773059166528912584522880000000000
1315447237773911946134425272753289479881991747923583177050255642481674704628704931514760179206953751034779825021930467488048287527948348080445051638640410578176183273686210152407875387233574666443500111746616434509656615544017913397500813030451524623897498205702352484426321970200000000000
143243919932521508681778540783805162083444879782412778783772991487171782942105530816273865339492686844402533802807617383061937682957745666521017919547189216882289724428339289804106242974679856331676933419038687780082133763079364597601152331375551295507841915476374119712859146472476491630296514727830236622514339341062000000000000
Technical details
Node id
n-a81229ffca43594f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-fa8d58a7053642bf

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:           21,             36;
  2:          441,           1512,            729;
  3:         9261,          47628,          45927,          11664;
  4:       194481,        1333584,        1928934,         979776,         164025;
  5:      4084101,       35006580,       67512690,       51438240,       17222625,        2125764;
  6:     85766121,      882165816,     2126649735,     2160406080,     1085025375,      267846264,       26040609;
  7:   1801088541,    21613062492,    62523502209,    79394923440,    53166243375,    19686700404,     3827969523,     306110016;
  8:  37822859361,   518713499808,  1750658061852,  2667669427584,  2232982221750,  1102455222624,   321549439932,   51426482688,   3486784401;
  9: 794280046581, 12254606432964, 47267767670004, 84031586968896, 84406727982150, 52091009268984, 20257614715716, 4859802614016, 659002251789, 38742048900;
%F Formula
The row polynomials satisfy (49+(63*n-21)*x+9*n^2*x^2)*P(n,x) = (1029+(1323*n+1323)*x+(189*n^2+945*n+567)*x^2+(81*n^2+162*n+81)*x^3)*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[((1029 + (1323*n + 1323)*x + (189*n^2 + 945*n + 567)*x^2 + (81*n^2 + 162*n + 81)*x^3)*P[n - 1, x])/(49 + (63*n - 21)*x + 9*n^2*x^2)]];
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] = ((1029 + (1323*n + 1323)*x + (189*n^2 + 945*n + 567)*x^2 + (81*n^2 + 162*n + 81)*x^3)*v[i - 1])/(49 + (63*n - 21)*x + 9*n^2*x^2)); 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.