Real-rooted atlas

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 n uniform [-1, 1] variables is between k and k+1.

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + x + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 3\,x^{2} - x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0213
18126127
24812011654121351
33841344195215206801722216075
4384017280336003696025232110003020490401131463
546080253440618240881280813248507808217392630801192013447413413907
664512041932801231104021584640251623682053497612018848507640015328323219444454836541401103429791
710321920774144002657894405535129607807242247884011525871887363271749121367619844255507296626881543248162456997227013581223435

All 15 rows

Technical details
Node id
n-e330d721682f502b · revision 0 · JSON
Definition source
derived:shift_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:          2,           1;
  1:          8,          12,            6,            1;
  2:         48,         120,          116,           54,           12,             1;
  3:        384,        1344,         1952,         1520,          680,           172,            22,             1;
  4:       3840,       17280,        33600,        36960,        25232,         11000,          3020,           490,           40,            1;
  5:      46080,      253440,       618240,       881280,       813248,        507808,        217392,         63080,        11920,         1344,           74,           1;
  6:     645120,     4193280,     12311040,     21584640,     25162368,      20534976,      12018848,       5076400,      1532832,       321944,        44548,        3654,         140,          1;
  7:   10321920,    77414400,    265789440,    553512960,    780724224,     788401152,     587188736,     327174912,    136761984,     42555072,      9662688,     1543248,      162456,       9972,       270,        1;
  8:  185794560,  1579253760,   6208634880,  14979686400,  24812605440,   29898086400,   27083043840,   18788943360,  10071510272,   4175432064,   1330100928,   320700576,    57001392,    7149096,    587292,    27474,     528,     1;
  9: 3715891200, 35300966400, 156377088000, 429030604800, 816520642560, 1143991296000, 1221798727680, 1015986616320, 665905660928, 345958874624, 142469423360, 46248193920, 11695604160, 2259010848, 323011824, 32534280, 2123040, 76600, 1042, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+2+(4*n+1)*x+2*n*x^2)*P(n-1,x) - (2*x+3*x^2+x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + x.
%t Mathematica
P[0, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(2*n + 2 + (4*n + 1)*x + 2*n*x^2)*P[n - 1, x] - (2*x + 3*x^2 + x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 2 + (4*n + 1)*x + 2*n*x^2)*v[i - 1] - (2*x + 3*x^2 + x^3)*deriv(v[i - 1], 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

Same fingerprint ?

References

  1. [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