mobius_x_over_x_plus_one of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and k+1.

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(1 + 7\,x + 19\,x^{2} + 25\,x^{3} + 16\,x^{4} + 4\,x^{5} - 2\,n x - 8\,n x^{2} - 12\,n x^{3} - 8\,n x^{4} - 2\,n x^{5}\right)\,P_{n-1}(x) + \left(x + 6\,x^{2} + 14\,x^{3} + 16\,x^{4} + 9\,x^{5} + 2\,x^{6}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	1	2															3
1	1	6	12	8													27
2	1	12	54	116	120	48											351
3	1	22	172	680	1520	1952	1344	384									6075
4	1	40	490	3020	11000	25232	36960	33600	17280	3840							131463
5	1	74	1344	11920	63080	217392	507808	813248	881280	618240	253440	46080					3413907
6	1	140	3654	44548	321944	1532832	5076400	12018848	20534976	25162368	21584640	12311040	4193280	645120			103429791
7	1	270	9972	162456	1543248	9662688	42555072	136761984	327174912	587188736	788401152	780724224	553512960	265789440	77414400	10321920	3581223435

All 15 rows

Technical details

Node id: n-b38b95ff2228e17d · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-8569cda4d5c26e17

Observed on the stored rows ?

Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, intercept 1)
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,    2;
  1: 1,    6,    12,       8;
  2: 1,   12,    54,     116,      120,        48;
  3: 1,   22,   172,     680,     1520,      1952,       1344,         384;
  4: 1,   40,   490,    3020,    11000,     25232,      36960,       33600,       17280,         3840;
  5: 1,   74,  1344,   11920,    63080,    217392,     507808,      813248,      881280,       618240,       253440,        46080;
  6: 1,  140,  3654,   44548,   321944,   1532832,    5076400,    12018848,    20534976,     25162368,     21584640,     12311040,       4193280,        645120;
  7: 1,  270,  9972,  162456,  1543248,   9662688,   42555072,   136761984,   327174912,    587188736,    788401152,    780724224,     553512960,     265789440,      77414400,     10321920;
  8: 1,  528, 27474,  587292,  7149096,  57001392,  320700576,  1330100928,  4175432064,  10071510272,  18788943360,  27083043840,   29898086400,   24812605440,   14979686400,   6208634880,   1579253760,    185794560;
  9: 1, 1042, 76600, 2123040, 32534280, 323011824, 2259010848, 11695604160, 46248193920, 142469423360, 345958874624, 665905660928, 1015986616320, 1221798727680, 1143991296000, 816520642560, 429030604800, 156377088000, 35300966400, 3715891200;

%F Formula

The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1-(2*n-7)*x-(8*n-19)*x^2-(12*n-25)*x^3-(8*n-16)*x^4-(2*n-4)*x^5)*P(n-1,x) + (x+6*x^2+14*x^3+16*x^4+9*x^5+2*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((1 - (2*n - 7)*x - (8*n - 19)*x^2 - (12*n - 25)*x^3 - (8*n - 16)*x^4 - (2*n - 4)*x^5)*P[n - 1, x] + (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + 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 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((1 - (2*n - 7)*x - (8*n - 19)*x^2 - (12*n - 25)*x^3 - (8*n - 16)*x^4 - (2*n - 4)*x^5)*v[i - 1] + (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + 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), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] v…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] v…?
    
    Product of real-rooted factors (elementary) polytool ? step ›
    
    from row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum o…?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum o…?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
  - ↑ Product of real-rooted factors (elementary) (above)

Related ?

Same fingerprint ?

shift_x_plus_one of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of…

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