Real-rooted atlas

A038255 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 - 28\,x + 84\,n x + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(343 + 588\,x + 336\,x^{2} + 64\,x^{3} + 588\,n x + 560\,n x^{2} + 128\,n x^{3} + 112\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
171623
249224144417
33432352302410246743
424012195242336286726400101761
516807192080493920501760224000368641465431
61176491613472518616070246404704000154828820070420394913
7823543131766885082436886051840768320003793305698344961048576276524567
8576480110541350447436076896378060810756480007080837122753658885872025653084163672445953
9403536078301313444269246912101196963841355316480011152318464578268364818496880643344302082621440047957927831
1028247524964565771203735591048010119696384015812025600015613245849610119696384043159388160117050572801835008000126877696617567936161
111977326743497156438243196005674409740207769601739322816000200369988403215584332431368308182220803004298035207064780800097695825926039797767859039654103
121384128720137964673465626846447664969090860584960182628895680002404439860838421818065403904139577461309446309025873920197813862400041032246886450734301184283534950499003149702017
13968890104072878987737808222093339773768272683132313618465810563200027350503417036828363485025075221169248298598411482427090534445002653696000124464482222082308410703872258016804864131533373441236255468757463
146782230728492170313833116818137622748190473702086087884818096494351936002978165927632896347452691557171229636947618037761875463091453952882052012441600304937981444096754080829931521264282343833612890270597126039797760015318668906775201
Technical details
Node id
n-6266b8f8b17b926c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-18b5ed15f2b69ef1

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:        7,        16;
  2:       49,       224,        144;
  3:      343,      2352,       3024,        1024;
  4:     2401,     21952,      42336,       28672,        6400;
  5:    16807,    192080,     493920,      501760,      224000,       36864;
  6:   117649,   1613472,    5186160,     7024640,     4704000,     1548288,     200704;
  7:   823543,  13176688,   50824368,    86051840,    76832000,    37933056,    9834496,    1048576;
  8:  5764801, 105413504,  474360768,   963780608,  1075648000,   708083712,  275365888,   58720256,   5308416;
  9: 40353607, 830131344, 4269246912, 10119696384, 13553164800, 11152318464, 5782683648, 1849688064, 334430208, 26214400;
%F Formula
The row polynomials satisfy (49+(84*n-28)*x+16*n^2*x^2)*P(n,x) = (343+(588*n+588)*x+(112*n^2+560*n+336)*x^2+(64*n^2+128*n+64)*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[((343 + (588*n + 588)*x + (112*n^2 + 560*n + 336)*x^2 + (64*n^2 + 128*n + 64)*x^3)*P[n - 1, x])/(49 + (84*n - 28)*x + 16*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] = ((343 + (588*n + 588)*x + (112*n^2 + 560*n + 336)*x^2 + (64*n^2 + 128*n + 64)*x^3)*v[i - 1])/(49 + (84*n - 28)*x + 16*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.