A370280 -id:A370280 - cp's OEIS frontend

A370282 Coefficient of x^n in the expansion of 1/( (1-x)^3 - x )^n.

1, 4, 42, 499, 6250, 80634, 1060269, 14127852, 190102482, 2577310285, 35150819132, 481734467955, 6628611532621, 91517611501008, 1267182734325900, 17589579427715124, 244689432718144770, 3410399867585709501, 47613678409439712861, 665756829352248572725

Offset: 0

Seiichi Manyama, Feb 13 2024

nonn

Cf. A069723, A370280.

Cf. A369215.

a(n) = sum(k=0, n, binomial(n+k-1, k)*binomial(4*n+2*k-1, n-k));

a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(4*n+2*k-1,n-k).

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * ((1-x)^3 - x) ). See A369215.

A124040 Triangle read by rows: characteristic polynomials of certain matrices, see Mathematica program.

Original entry on oeis.org

3, 3, -1, 8, -6, 1, 20, -24, 9, -1, 45, -84, 50, -12, 1, 125, -275, 225, -85, 15, -1, 320, -864, 900, -468, 129, -18, 1, 845, -2639, 3339, -2219, 840, -182, 21, -1, 2205, -7896, 11756, -9528, 4610, -1368, 244, -24, 1, 5780, -23256, 39825, -38121, 22518, -8532, 2079, -315, 27, -1, 15125, -67650, 130975, -144660, 101065, -46746, 14525, -3000, 395, -30, 1

Offset: 1

Gary W. Adamson and Roger L. Bagula, Nov 04 2006

Matrices:
X 1 : {{3}},
X 2 : {{3, 1}, {1, 3}},
X 3 : {{3, 1, 1}, {1, 3, 1}, {1, 1, 3}},
X 4 : {{3, 1, 0, 1}, {1, 3, 1, 0}, {0, 1, 3, 1}, {1, 0, 1, 3}},
X 5 : {{3, 1, 0, 0, 1}, {1, 3, 1, 0, 0}, {0, 1, 3, 1, 0}, {0, 0, 1, 3, 1}, {1, 0, 0, 1, 3}}.

			Triangle begins:
     3;
     3,     -1;
     8,     -6,     1;
    20,    -24,     9,     -1;
    45,    -84,    50,    -12,     1;
   125,   -275,   225,    -85,    15,    -1;
   320,   -864,   900,   -468,   129,   -18,    1;
   845,  -2639,  3339,  -2219,   840,  -182,   21,   -1;
  2205,  -7896, 11756,  -9528,  4610, -1368,  244,  -24,  1;
  5780, -23256, 39825, -38121, 22518, -8532, 2079, -315, 27, -1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A010675, A099921, A201630, A370280.

T[n_, m_, d_]:= If[n==m, 3, If[n==m-1 || n==m+1, 1, If[(n==1 && m==d) || (n==d && m==1), 1, 0]]];
M[d_]:= Table[T[n, m, d], {n,d}, {m,d}];
Join[{M[1]}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d,12}]]

From G. C. Greubel, Feb 03 2025: (Start)
T(n, 1) = A099921(n-1) + 3*[n=1] - 2*[n=2] + 3*[n=3].
T(n, 2) = -(n-1)*Fibonacci(2*n-2).
T(n, 3) = (1/10)*(n-1)*(2*(n-1)*Fibonacci(2*n-1) - (n+2)*Fibonacci(2*n-2)).
T(n, 4) = (1/150)*(n-1)*(18*(n-1)*Fibonacci(2*n-1) - (5*n^2 - n + 18)*Fibonacci(2*n-2)).
T(n, 5) = (1/600)*(n-1)*(2*(n-1)*(n^2-2*n+24)*Fibonacci(2*n-1) - (n^3+15*n^2 -10*n+48)*Fibonacci(2*n-2)).
T(n, n) = (-1)^(n-1) + 2*[n=1].
T(n, n-1) = 3*(-1)^n*(n-1).
T(n, n-2) = (1/2)*(-1)^(n+1)*(n-1)*(9*n-20) + [n=3].
T(n, n-3) = (3/2)*(-1)^n*(n-1)*(n-3)*(3*n-8) + 2*[n=4].
T(n, n-4) = (1/8)*(-1)^(n-1)*n*(n-3)*(27*n^2-117*n+130) - 2*[n=5].
T(n, n-5) = (3/40)*(-1)^n*(n-1)*(n-4)*(n-5)*(27*n^2-195*n+362) + 2*[n=6].
T(n, n-6) = (1/240)*(-1)^(n-1)*(n-1)*(n-5)*(n-6)(243*n^3-2997*n^2+12528*n -17752) -2*[n=7].
T(2*n-1, n) = 2*(-1)^(n-1)*A370280(n-1) + [n=1].
Sum_{k=1..n} T(n, k) = A010675(n-1) + 3*[n=1] -2*[n=2] +3*[n=3].
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A201630(n-2) - A201630(n-1) + (1/4)*[n=1] + (7/2)*[n=2] + 2*[n=3].
(End)

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A370282 Coefficient of x^n in the expansion of 1/( (1-x)^3 - x )^n.

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A124040 Triangle read by rows: characteristic polynomials of certain matrices, see Mathematica program.

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