Real-rooted atlas

A371259 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-500 + 155\,x - 6\,x^{2} - x^{3} - 425\,n + 111\,n x - 6\,n x^{2} - 135\,n^{2} + 26\,n^{2} x - n^{2} x^{2} - 19\,n^{3} + 2\,n^{3} x - n^{4}\right)\,P_{n-2}(x) + \left(-220\,x + 89\,x^{2} - 9\,x^{3} - 139\,n x + 38\,n x^{2} - 2\,n x^{3} - 29\,n^{2} x + 4\,n^{2} x^{2} - 2\,n^{3} x\right)\,P_{n-2}'(x) + \left(-20\,x^{2} + 9\,x^{3} - x^{4} - 9\,n x^{2} + 2\,n x^{3} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(56 - 10\,x + x^{2} + 21\,n - 2\,n x + 2\,n^{2}\right)\,P_{n-1}(x) + \left(13\,x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 37 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
137138
2186410211967
3122760102642031133228
410243872116742435096348111446741
5105661699215416841662811129436054511217161426
6132034779648236992538881229149056249305122176808021156988331587
71965829289011242210678804482678639777286889088768810016324504641127124154195290280

All 15 rows

Technical details
Node id
n-56990ef96985b6f3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-71f59c90b30e1bfc

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:                 37,                  1;
  2:               1864,                102,                 1;
  3:             122760,              10264,               203,               1;
  4:           10243872,            1167424,             35096,             348,              1;
  5:         1056616992,          154168416,           6281112,           94360,            545,            1;
  6:       132034779648,        23699253888,        1229149056,        24930512,         217680,          802,         1;
  7:     19658292890112,      4221067880448,      267863977728,      6889088768,       81001632,       450464,      1127,       1;
  8:   3438754166679552,    865104623294976,    65349982987776,   2048444003840,    30454998528,    227644352,    858704,    1528,    1;
  9: 698322321862502400, 202525804487374848, 17845433463545856, 663389285842944, 11986378599680, 113055612544, 572410256, 1534416, 2013, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4+19*n^3+135*n^2+425*n+500-(2*n^3+26*n^2+111*n+155)*x+(n^2+6*n+6)*x^2+x^3)*P(n-2,x) - ((2*n^3+29*n^2+139*n+220)*x-(4*n^2+38*n+89)*x^2+(2*n+9)*x^3)*d/dx P(n-2,x) - ((n^2+9*n+20)*x^2-(2*n+9)*x^3+x^4)*d^2/dx^2 P(n-2,x) + (2*n^2+21*n+56-(2*n+10)*x+x^2)*P(n-1,x) + (2*n+13)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 37 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 37 + x;
P[n_, x_] := P[n, x] = Expand[-(n^4 + 19*n^3 + 135*n^2 + 425*n + 500 - (2*n^3 + 26*n^2 + 111*n + 155)*x + (n^2 + 6*n + 6)*x^2 + x^3)*P[n - 2, x] - ((2*n^3 + 29*n^2 + 139*n + 220)*x - (4*n^2 + 38*n + 89)*x^2 + (2*n + 9)*x^3)*D[P[n - 2, x], x] - ((n^2 + 9*n + 20)*x^2 - (2*n + 9)*x^3 + x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 21*n + 56 - (2*n + 10)*x + x^2)*P[n - 1, x] + (2*n + 13)*x*D[P[n - 1, x], x] + x^2*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; v[2] = 37 + x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 19*n^3 + 135*n^2 + 425*n + 500 - (2*n^3 + 26*n^2 + 111*n + 155)*x + (n^2 + 6*n + 6)*x^2 + x^3)*v[i - 2] - ((2*n^3 + 29*n^2 + 139*n + 220)*x - (4*n^2 + 38*n + 89)*x^2 + (2*n + 9)*x^3)*deriv(v[i - 2], x) - ((n^2 + 9*n + 20)*x^2 - (2*n + 9)*x^3 + x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 21*n + 56 - (2*n + 10)*x + x^2)*v[i - 1] + (2*n + 13)*x*deriv(v[i - 1], x) + x^2*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

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.