Real-rooted atlas

A371081 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(9 + 35\,x + 52\,x^{2} + 37\,x^{3} + 13\,x^{4} + 2\,x^{5} + 6\,n + 13\,n x - 18\,n x^{3} - 14\,n x^{4} - 3\,n x^{5} + n^{2} + n^{2} x - 2\,n^{2} x^{2} - 2\,n^{2} x^{3} + n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}(x) + \left(7\,x + 32\,x^{2} + 58\,x^{3} + 52\,x^{4} + 23\,x^{5} + 4\,x^{6} + 2\,n x + 6\,n x^{2} + 4\,n x^{3} - 4\,n x^{4} - 6\,n x^{5} - 2\,n x^{6}\right)\,P_{n-1}'(x) + \left(x^{2} + 5\,x^{3} + 10\,x^{4} + 10\,x^{5} + 5\,x^{6} + x^{7}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1161733
24008524531705
314400461484921217465127225
470560030254724855160345652092123312963985
545158400242946432521639200558861488298799464637891451731194129
6365783040023707584000638615525449153395676873641038360315354795485616599013293554040633
736578304000027771719040009010864935936162005229902721743450072704011232879103268401257345163661314815707361647444309225

All 15 rows

Technical details
Node id
n-50b777e2c50736e8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9558912bc02a9014

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:               16,                17;
  2:              400,               852,                453;
  3:            14400,             46148,              49212,              17465;
  4:           705600,           3025472,            4855160,            3456520,              921233;
  5:         45158400,         242946432,          521639200,          558861488,           298799464,            63789145;
  6:       3657830400,       23707584000,        63861552544,        91533956768,         73641038360,         31535479548,         5616599013;
  7:     365783040000,     2777171904000,      9010864935936,     16200522990272,      17434500727040,      11232879103268,      4012573451636,       613148157073;
  8:   44259747840000,   385647215616000,   1465422132893184,   3172785038051072,    4282070715283392,    3689768554863360,   1982718348877776,    607568974811920,     81298838448001;
  9: 6373403688960000, 62740093307904000, 273522037261400064, 693369747801522432, 1126654236733427968, 1217238635615835840, 874615142613135424, 403092467127352960, 108147204453430848, 12871080897739073;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (n^2+6*n+9+(n^2+13*n+35)*x-(2*n^2-52)*x^2-(2*n^2+18*n-37)*x^3+(n^2-14*n+13)*x^4+(n^2-3*n+2)*x^5)*P(n-1,x) + ((2*n+7)*x+(6*n+32)*x^2+(4*n+58)*x^3-(4*n-52)*x^4-(6*n-23)*x^5-(2*n-4)*x^6)*d/dx P(n-1,x) + (x^2+5*x^3+10*x^4+10*x^5+5*x^6+x^7)*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[Together[((n^2 + 6*n + 9 + (n^2 + 13*n + 35)*x - (2*n^2 - 52)*x^2 - (2*n^2 + 18*n - 37)*x^3 + (n^2 - 14*n + 13)*x^4 + (n^2 - 3*n + 2)*x^5)*P[n - 1, x] + ((2*n + 7)*x + (6*n + 32)*x^2 + (4*n + 58)*x^3 - (4*n - 52)*x^4 - (6*n - 23)*x^5 - (2*n - 4)*x^6)*D[P[n - 1, x], x] + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*D[P[n - 1, x], {x, 2}])/(1 + 2*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] = ((n^2 + 6*n + 9 + (n^2 + 13*n + 35)*x - (2*n^2 - 52)*x^2 - (2*n^2 + 18*n - 37)*x^3 + (n^2 - 14*n + 13)*x^4 + (n^2 - 3*n + 2)*x^5)*v[i - 1] + ((2*n + 7)*x + (6*n + 32)*x^2 + (4*n + 58)*x^3 - (4*n - 52)*x^4 - (6*n - 23)*x^5 - (2*n - 4)*x^6)*deriv(v[i - 1], x) + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*deriv(deriv(v[i - 1], x), x))/(1 + 2*x + 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. [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.