Real-rooted atlas

A147716 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 5\,x + 15\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(375 + 225\,x + 45\,x^{2} + 3\,x^{3} + 225\,n x + 75\,n x^{2} + 6\,n x^{3} + 15\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1151227
222536081666
333758100364543215552
450625162000109350259202025349920
57593753037500273375097200015187587487663248
611390625546750006150937529160000683437578732035721164392416
717085937595681250012916968757654500002392031254133430037507051399683469246848

All 15 rows

Technical details
Node id
n-ed584f5c1991eac3 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7fe862d483d8a327

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,           12;
  2:         225,          360,           81;
  3:        3375,         8100,         3645,          432;
  4:       50625,       162000,       109350,        25920,         2025;
  5:      759375,      3037500,      2733750,       972000,       151875,        8748;
  6:    11390625,     54675000,     61509375,     29160000,      6834375,      787320,       35721;
  7:   170859375,    956812500,   1291696875,    765450000,    239203125,    41334300,     3750705,     139968;
  8:  2562890625,  16402500000,  25833937500,  18370800000,   7176093750,  1653372000,   225042300,   16796160,   531441;
  9: 38443359375, 276792187500, 498225937500, 413343000000, 193754531250, 55801305000, 10126903500, 1133740800, 71744535, 1968300;
%F Formula
The row polynomials satisfy (25+(15*n-5)*x+n^2*x^2)*P(n,x) = (375+(225*n+225)*x+(15*n^2+75*n+45)*x^2+(3*n^2+6*n+3)*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[((375 + (225*n + 225)*x + (15*n^2 + 75*n + 45)*x^2 + (3*n^2 + 6*n + 3)*x^3)*P[n - 1, x])/(25 + (15*n - 5)*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] = ((375 + (225*n + 225)*x + (15*n^2 + 75*n + 45)*x^2 + (3*n^2 + 6*n + 3)*x^3)*v[i - 1])/(25 + (15*n - 5)*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

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.