Real-rooted atlas

A027467 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(225 - 15\,x + 45\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(3375 + 675\,x + 45\,x^{2} + x^{3} + 675\,n x + 75\,n x^{2} + 2\,n x^{3} + 15\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115419
22251209354
333752700405166496
450625540001215096025117760
57593751012500303750360001875362113536
61139062518225000683437510800008437532404937617664
7170859375318937500143521875283500002953125170100514564664797184
825628906255467500000287043750068040000088593750680400030870076808111676942336
93844335937592264062500553584375001530900000023920312502296350001389150051840010935100204010946560
105766503906251537734375000103797070312532805000000059800781250688905000052093125025920000820125150001213547642986496
1186497558593752537261718750019029462890625676603125000014095898437501894488750001719073125010692000004510687512375001996514461435212201984
12129746337890625415188281250000342530332031250135320625000000317157714843754871542500000515721937500384912000002029809375742500001796850259201691059929209176064
13194619506835937567468095703125006072128613281250263875218750000068717504882812511874384843750014366539687500125096400000079162565625361968750011679525025272003295519618225504742014976
1429192926025390625108986923828125000106262250732421875503761781250000001443067602539062527706897968750003771216667968753752892000000027706897968751520268750006131750625176904000346027541160225312437224148828160
Technical details
Node id
n-45961366af1d7d40 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7e2ca71e44b89f5f

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:          15,           4;
  2:         225,         120,           9;
  3:        3375,        2700,         405,          16;
  4:       50625,       54000,       12150,         960,         25;
  5:      759375,     1012500,      303750,       36000,       1875,        36;
  6:    11390625,    18225000,     6834375,     1080000,      84375,      3240,       49;
  7:   170859375,   318937500,   143521875,    28350000,    2953125,    170100,     5145,     64;
  8:  2562890625,  5467500000,  2870437500,   680400000,   88593750,   6804000,   308700,   7680,    81;
  9: 38443359375, 92264062500, 55358437500, 15309000000, 2392031250, 229635000, 13891500, 518400, 10935, 100;
%F Formula
The row polynomials satisfy (225+(45*n-15)*x+n^2*x^2)*P(n,x) = (3375+(675*n+675)*x+(15*n^2+75*n+45)*x^2+(n^2+2*n+1)*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[((3375 + (675*n + 675)*x + (15*n^2 + 75*n + 45)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(225 + (45*n - 15)*x + 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] = ((3375 + (675*n + 675)*x + (15*n^2 + 75*n + 45)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(225 + (45*n - 15)*x + 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

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.