A336541 Expansion of (2*x)/(sqrt((1-x)^2-4*x^2*y)+3*x-1).

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 10, 16, 7, 1, 0, 1, 15, 40, 30, 9, 1, 0, 1, 21, 85, 100, 48, 11, 1, 0, 1, 28, 161, 280, 196, 70, 13, 1, 0, 1, 36, 280, 686, 672, 336, 96, 15, 1, 0, 1, 45, 456, 1512, 2016, 1344, 528, 126, 17, 1

Offset

0,9

Links

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)

Formula

T(n,m) = C(n-1,n-m)*Sum_{k=0..n} C(n-m,m-k)*k/m, T(0,0)=1, T(0,m)=0, m>0.

Sum_{m=0..n} T(n,m) = A005773(n).

m*T(m, n) + (-2*m + 6)*T(m - 3, n) + (4*m - 12)*T(m - 3, n - 2) + (7*m - 21)*T(m - 3, n - 1) + (5*m - 9)*T(m - 2, n) + (-2*m + 9)*T(m - 2, n - 1) + (-4*m + 3)*T(m - 1, n) - m*T(m - 1, n - 1)= 0 for m>=3, n>=2. - Robert Israel, Oct 05 2020

Example

1,
0,1,
0,1, 1,
0,1, 3, 1,
0,1, 6, 5,  1,
0,1,10,16,  7, 1,
0,1,15,40, 30, 9, 1,
0,1,21,85,100,48,11,1

Maple

T:= proc(m, n) option remember;
if m < n then 0
elif m = n then 1
elif n=0 then 0
elif n=1 then 1
else
(-(-2*m + 6)*procname(m - 3, n) - (4*m - 12)*procname(m - 3, n - 2) - (7*m - 21)*procname(m - 3, n - 1) - (5*m - 9)*procname(m - 2, n) - (-2*m + 9)*procname(m - 2, n - 1) - (-4*m + 3)*procname(m - 1, n) + m*procname(m - 1, n - 1))/m
fi
end proc:
seq(seq(T(n, m), m=0..n), n=0..20); # Robert Israel, Oct 05 2020

Mathematica

T[n_, m_] := If[n == m, 1, If[m == 0, 0, Binomial[n-1, n-m]*
   Sum[Binomial[n-m, m-k]*k, {k, 0, n}]/m]];
Table[T[n, m], {n, 0, 20}, {m, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2023 *)

Prog

(Maxima)
T(n, m):=if m=0 and n=0 then 1 else if m=0 then 0 else (binomial(n-1, n-m)*sum(binomial(n-m, m-k)*k, k, 0, n))/m;

Crossrefs

Cf. A005773.

Sequence in context: A098157 A293617 A165253 * A317659 A059045 A348210

Adjacent sequences: A336538 A336539 A336540 * A336542 A336543 A336544

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Oct 04 2020

Status

approved