A160562 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 12\,x + 37\,x^{2} + 50\,x^{3} + 32\,x^{4} + 8\,x^{5} - 8\,n x - 36\,n x^{2} - 60\,n x^{3} - 44\,n x^{4} - 12\,n x^{5} + 4\,n^{2} x^{2} + 12\,n^{2} x^{3} + 12\,n^{2} x^{4} + 4\,n^{2} x^{5}\right)\,P_{n-1}(x) + \left(8\,x + 48\,x^{2} + 112\,x^{3} + 128\,x^{4} + 72\,x^{5} + 16\,x^{6} - 8\,n x^{2} - 32\,n x^{3} - 48\,n x^{4} - 32\,n x^{5} - 8\,n x^{6}\right)\,P_{n-1}'(x) + \left(4\,x^{2} + 20\,x^{3} + 40\,x^{4} + 40\,x^{5} + 20\,x^{6} + 4\,x^{7}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	12	12						25
3	1	94	220	128					443
4	1	824	3432	4480	1872				10609
5	1	7386	54504	124288	114768	37600			338547
6	1	66436	963796	3464832	5294512	3717824	990784		14498185
7	1	597878	19444452	102311168	228525872	257240032	144512576	32333824	784965803

All 15 rows

Technical details

Node id: n-8272e45f524cffe5 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-44af577288c301d1

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,       12,          12;
  3: 1,       94,         220,          128;
  4: 1,      824,        3432,         4480,         1872;
  5: 1,     7386,       54504,       124288,       114768,        37600;
  6: 1,    66436,      963796,      3464832,      5294512,      3717824,        990784;
  7: 1,   597878,    19444452,    102311168,    228525872,    257240032,     144512576,     32333824;
  8: 1,  5380848,   434693904,   3199817984,   9824291808,  15812723456,   14083318528,   6597042176,   1272660224;
  9: 1, 48427570, 10318501264, 105798528256, 434258986976, 943222606912, 1187865671424, 875206156288, 351359873280, 59527313920;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1-(8*n-12)*x+(4*n^2-36*n+37)*x^2+(12*n^2-60*n+50)*x^3+(12*n^2-44*n+32)*x^4+(4*n^2-12*n+8)*x^5)*P(n-1,x) + (8*x-(8*n-48)*x^2-(32*n-112)*x^3-(48*n-128)*x^4-(32*n-72)*x^5-(8*n-16)*x^6)*d/dx P(n-1,x) + (4*x^2+20*x^3+40*x^4+40*x^5+20*x^6+4*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[((1 - (8*n - 12)*x + (4*n^2 - 36*n + 37)*x^2 + (12*n^2 - 60*n + 50)*x^3 + (12*n^2 - 44*n + 32)*x^4 + (4*n^2 - 12*n + 8)*x^5)*P[n - 1, x] + (8*x - (8*n - 48)*x^2 - (32*n - 112)*x^3 - (48*n - 128)*x^4 - (32*n - 72)*x^5 - (8*n - 16)*x^6)*D[P[n - 1, x], x] + (4*x^2 + 20*x^3 + 40*x^4 + 40*x^5 + 20*x^6 + 4*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] = ((1 - (8*n - 12)*x + (4*n^2 - 36*n + 37)*x^2 + (12*n^2 - 60*n + 50)*x^3 + (12*n^2 - 44*n + 32)*x^4 + (4*n^2 - 12*n + 8)*x^5)*v[i - 1] + (8*x - (8*n - 48)*x^2 - (32*n - 112)*x^3 - (48*n - 128)*x^4 - (32*n - 72)*x^5 - (8*n - 16)*x^6)*deriv(v[i - 1], x) + (4*x^2 + 20*x^3 + 40*x^4 + 40*x^5 + 20*x^6 + 4*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

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 of scaled central factorial numbers, T(n,k) = A008958(n,n-k).?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[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
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.