Real-rooted atlas

A151775: Triangle read by rows: T(n,k) = value of (d^2n/dx^2n) (tan^(2k)(x)/cos(x)) at the point x = 0.

A151775 ↗ nonnegative coefficients

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

Recurrence from A151775

\(P_{n}(x) = \left(1 + 2\,x\right)\,P_{n-1}(x) + \left(2 + 12\,x + 10\,x^{2}\right)\,P_{n-1}'(x) + \left(4\,x + 8\,x^{2} + 4\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
25282457
36166213207202763
41385245688366410080040320250737
550521132612267490401333584011491200362880036581523
62702765983291086926993041979524800273962304018361728004790016007828053417
7199360981959607558288966364760342459477360673354281600715588473600392302310400871782912002309644635483

All 15 rows b-files: A000281 (row sums), A151775 ?

Technical details
Node id
n-3338feb205e3b77f · revision 0 · JSON
Definition source
oeis_catalog:A151775
Fingerprint
f6-a766906061ecf2ab

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 ?

  • A151775 triangle 30 terms, 7 rows · tabl · nonn · catalog b-file?

Row sums ? A000281

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             1;
  1:             1,               2;
  2:             5,              28,               24;
  3:            61,             662,             1320,               720;
  4:          1385,           24568,            83664,            100800,             40320;
  5:         50521,         1326122,          6749040,          13335840,          11491200,            3628800;
  6:       2702765,        98329108,        692699304,        1979524800,        2739623040,         1836172800,          479001600;
  7:     199360981,      9596075582,      88966364760,      342459477360,      673354281600,       715588473600,       392302310400,       87178291200;
  8:   19391512145,   1192744081648,   14036558182944,    69485886288000,   181217064188160,    270530842982400,    233114750668800,   108101081088000,    20922789888000;
  9: 2404879675441, 183983154281042, 2674388403120480, 16475552674505280, 54515459807827200, 106867999164326400, 128349384991027200, 92958909332889600, 37347179950080000, 6402373705728000;
%F Formula
The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) + (2+12*x+10*x^2)*d/dx P(n-1,x) + (4*x+8*x^2+4*x^3)*d^2/dx^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[(1 + 2*x)*P[n - 1, x] + (2 + 12*x + 10*x^2)*D[P[n - 1, x], x] + (4*x + 8*x^2 + 4*x^3)*D[P[n - 1, x], {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] = (1 + 2*x)*v[i - 1] + (2 + 12*x + 10*x^2)*deriv(v[i - 1], x) + (4*x + 8*x^2 + 4*x^3)*deriv(deriv(v[i - 1], x), 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

A151775 triangle, offset 0, from P0
%e Example
Triangle begins:
  0:             1;
  1:             1,               2;
  2:             5,              28,               24;
  3:            61,             662,             1320,               720;
  4:          1385,           24568,            83664,            100800,             40320;
  5:         50521,         1326122,          6749040,          13335840,          11491200,            3628800;
  6:       2702765,        98329108,        692699304,        1979524800,        2739623040,         1836172800,          479001600;
  7:     199360981,      9596075582,      88966364760,      342459477360,      673354281600,       715588473600,       392302310400,       87178291200;
  8:   19391512145,   1192744081648,   14036558182944,    69485886288000,   181217064188160,    270530842982400,    233114750668800,   108101081088000,    20922789888000;
  9: 2404879675441, 183983154281042, 2674388403120480, 16475552674505280, 54515459807827200, 106867999164326400, 128349384991027200, 92958909332889600, 37347179950080000, 6402373705728000;
%F Formula
The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) + (2+12*x+10*x^2)*d/dx P(n-1,x) + (4*x+8*x^2+4*x^3)*d^2/dx^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[(1 + 2*x)*P[n - 1, x] + (2 + 12*x + 10*x^2)*D[P[n - 1, x], x] + (4*x + 8*x^2 + 4*x^3)*D[P[n - 1, x], {x, 2}]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (1 + 2*x)*v[i - 1] + (2 + 12*x + 10*x^2)*deriv(v[i - 1], x) + (4*x + 8*x^2 + 4*x^3)*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.