A102413 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 3\,x - 3\,x^{2} - x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(5 + 39\,x + 123\,x^{2} + 201\,x^{3} + 180\,x^{4} + 84\,x^{5} + 16\,x^{6} - 8\,n - 59\,n x - 181\,n x^{2} - 293\,n x^{3} - 263\,n x^{4} - 124\,n x^{5} - 24\,n x^{6} + 3\,n^{2} + 20\,n^{2} x + 58\,n^{2} x^{2} + 92\,n^{2} x^{3} + 83\,n^{2} x^{4} + 40\,n^{2} x^{5} + 8\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(2 + 31\,x + 156\,x^{2} + 386\,x^{3} + 534\,x^{4} + 423\,x^{5} + 180\,x^{6} + 32\,x^{7} - n - 14\,n x - 70\,n x^{2} - 176\,n x^{3} - 249\,n x^{4} - 202\,n x^{5} - 88\,n x^{6} - 16\,n x^{7}\right)\,P_{n-1}'(x) + \left(x + 13\,x^{2} + 58\,x^{3} + 130\,x^{4} + 165\,x^{5} + 121\,x^{6} + 48\,x^{7} + 8\,x^{8}\right)\,P_{n-1}''(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 1 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	6	6						13
3	1	9	21	14					45
4	1	12	46	68	34				161
5	1	15	80	190	205	82			573
6	1	18	123	408	699	594	198		2041
7	1	21	175	749	1785	2387	1673	478	7269

All 15 rows

Technical details

Node id: n-9a90d366df7da363 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-65f5fe102fc0d40e

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: 1,  2;
  2: 1,  6,   6;
  3: 1,  9,  21,   14;
  4: 1, 12,  46,   68,   34;
  5: 1, 15,  80,  190,  205,    82;
  6: 1, 18, 123,  408,  699,   594,   198;
  7: 1, 21, 175,  749, 1785,  2387,  1673,   478;
  8: 1, 24, 236, 1240, 3814,  7064,  7740,  4616,  1154;
  9: 1, 27, 306, 1908, 7218, 17226, 26040, 24156, 12537, 2786;

%F Formula

The row polynomials satisfy (n-1+(3*n-3)*x+(3*n-3)*x^2+(n-1)*x^3)*P(n,x) = (3*n^2-8*n+5+(20*n^2-59*n+39)*x+(58*n^2-181*n+123)*x^2+(92*n^2-293*n+201)*x^3+(83*n^2-263*n+180)*x^4+(40*n^2-124*n+84)*x^5+(8*n^2-24*n+16)*x^6)*P(n-1,x) - (n-2+(14*n-31)*x+(70*n-156)*x^2+(176*n-386)*x^3+(249*n-534)*x^4+(202*n-423)*x^5+(88*n-180)*x^6+(16*n-32)*x^7)*d/dx P(n-1,x) + (x+13*x^2+58*x^3+130*x^4+165*x^5+121*x^6+48*x^7+8*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 8*n + 5 + (20*n^2 - 59*n + 39)*x + (58*n^2 - 181*n + 123)*x^2 + (92*n^2 - 293*n + 201)*x^3 + (83*n^2 - 263*n + 180)*x^4 + (40*n^2 - 124*n + 84)*x^5 + (8*n^2 - 24*n + 16)*x^6)*P[n - 1, x] - (n - 2 + (14*n - 31)*x + (70*n - 156)*x^2 + (176*n - 386)*x^3 + (249*n - 534)*x^4 + (202*n - 423)*x^5 + (88*n - 180)*x^6 + (16*n - 32)*x^7)*D[P[n - 1, x], x] + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*D[P[n - 1, x], {x, 2}])/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)]];
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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n^2 - 8*n + 5 + (20*n^2 - 59*n + 39)*x + (58*n^2 - 181*n + 123)*x^2 + (92*n^2 - 293*n + 201)*x^3 + (83*n^2 - 263*n + 180)*x^4 + (40*n^2 - 124*n + 84)*x^5 + (8*n^2 - 24*n + 16)*x^6)*v[i - 1] - (n - 2 + (14*n - 31)*x + (70*n - 156)*x^2 + (176*n - 386)*x^3 + (249*n - 534)*x^4 + (202*n - 423)*x^5 + (88*n - 180)*x^6 + (16*n - 32)*x^7)*deriv(v[i - 1], x) + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*deriv(deriv(v[i - 1], x), x))/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)); 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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

Related ?

Same fingerprint ?

A102413 → basis A007318

References

[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