A147563 - OEIS

%I #18 Mar 04 2023 02:02:04

%S 4,4,4,4,-2,4,16,-8,4,44,-6,-16,4,4,104,84,-136,34,4,228,606,-584,-24,

%T 102,-17,4,480,2832,-1088,-2208,1488,-248,4,988,11122,5536,-20840,

%U 8896,832,-992,124,4,2008,39772,74296,-118190,-2144,51952,-22112,2764

%N Irregular triangle, T(n, k) = [x^k] p(n, x), where p(n, x) = 4*Sum_{j=0..n} A008292(n+1, j) * (x/2)^j * (1-x/2)^(n-j), read by rows.

%H G. C. Greubel, <a href="/A147563/b147563.txt">Row n = 0..50 of the irregular triangle, flattened</a>

%F T(n, k) = coefficients [x^k]( p(n, x) ), where p(n, x) = 4*Sum_{j=0..n} A008292(n+1, j) * (x/2)^j * (1-x/2)^(n-j).

%F T(n, k) = round( (-1/2)^(k-2) * Sum_{j=0..k} (-1)^j*binomial(n-j, k-j) * A008292(n+1, j+1) ). - _G. C. Greubel_, Mar 03 2023

%e Irregular triangle begins as:

%e   4;

%e   4;

%e   4,    4,    -2;

%e   4,   16,    -8;

%e   4,   44,    -6,   -16,       4;

%e   4,  104,    84,  -136,      34;

%e   4,  228,   606,  -584,     -24,   102,   -17;

%e   4,  480,  2832, -1088,   -2208,  1488,  -248;

%e   4,  988, 11122,  5536,  -20840,  8896,   832,   -992,  124;

%e   4, 2008, 39772, 74296, -118190, -2144, 51952, -22112, 2764;

%t (* First program *)

%t nmax:= 15;

%t p[x_, n_]= (1-x)^(n+1)*PolyLog[-n, x]/x;

%t b= Table[CoefficientList[p[x, n], x], {n, nmax+1}];

%t F[n_]:= CoefficientList[4*Sum[b[[n+1]][[m+1]]*(x/2)^(n-m)*(1-x/2)^m, {m, 0, n}], x];

%t T[n_]:= If[IntegerQ[F[n]], F[n], Sign[F[n]]*Abs[Round[F[n] - 1/2]]];

%t Table[T[n], {n, 0, nmax}]//Flatten

%t (* Second program *)

%t A008292[n_, k_]:= Sum[(-1)^j*(k-j)^n*Binomial[n+1,j], {j,0,k}];

%t F[n_, k_]:= (-1/2)^(k-2)*Sum[(-1)^j*Binomial[n-j, k-j]*A008292[n+1, j+ 1], {j,0,k}];

%t T[n_, k_]:= If[IntegerQ[F[n,k]], F[n,k], Sign[F[n,k]]*Abs[Round[F[n, k] - 1/2]]];

%t Table[T[n, k], {n,0,16}, {k, 0, 2*Floor[n/2]}]//Flatten (* _G. C. Greubel_, Mar 03 2023 *)

%o (Magma)

%o A008292:= func< n,k | (&+[(-1)^j*Binomial(n+1, j)*(k-j)^n: j in [0..k]]) >;

%o T:= func< n,k | (-1/2)^(k-2)*(&+[(-1)^j*Binomial(n-j,k-j)*A008292(n+1,j+1): j in [0..k]]) >;

%o [Floor(T(n,k)): k in [0..2*Floor(n/2)], n in [0..16]]; // _G. C. Greubel_, Oct 27 2022; Mar 03 2023

%o (SageMath)

%o def A008292(n,k): return sum( (-1)^j*binomial(n+1, j)*(k-j)^n for j in range(k+1) )

%o def A147563(n,k): return floor((-1/2)^(k-2)*sum((-1)^j*binomial(n-j, k-j)*A008292(n+1,j+1) for j in range(k+1)))

%o flatten([[A147563(n,k) for k in range(2*floor(n/2) + 1)] for n in range(16)]) # _G. C. Greubel_, Oct 27 2022; Mar 03 2023

%Y Cf. A008292.

%K sign,tabf

%O 0,1

%A _Roger L. Bagula_, Nov 07 2008

%E Edited by _G. C. Greubel_, Oct 27 2022