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A169658 Triangle, read by rows, defined by T(n, k) = b(n, k) + b(n, n-k+1) - (b(n,1) + b(n,n)) + 1, where b(n, k) = (-1)^n*(n!/k!)^2 *binomial(n-1, k-1). 1
1, 1, 1, 1, 2, 1, 1, -96, -96, 1, 1, -98, 9602, -98, 1, 1, 129780, -365400, -365400, 129780, 1, 1, -12701092, 14791142, 23637602, 14791142, -12701092, 1, 1, 1219277248, -677310144, -1522967040, -1522967040, -677310144, 1219277248, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Row sums are: {1, 2, 4, -190, 9408, -471238, 27817704, -1961999870, 163293385984, -15674630045398, ...}.
LINKS
FORMULA
T(n, k) = b(n, k) + b(n, n-k+1) - b(n, n) - b(n, 1) + 1, where b(n, k) = (-1)^n*(n!/m!)^2 *binomial(n-1, k-1), where 1 <= k <= n, n >= 1.
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 2, 1;
1, -96, -96, 1;
1, -98, 9602, -98, 1;
1, 129780, -365400, -365400, 129780, 1;
1, -12701092, 14791142, 23637602, 14791142, -12701092, 1;
MATHEMATICA
L[n_, m_] = (-1)^n*(n!/m!)^2*Binomial[n-1, m-1];
t[n_, m_] = L[n, m] + L[n, n-m+1];
Table[t[n, m] - t[n, 1] + 1, {n, 1, 10}, {m, 1, n}]//Flatten
PROG
(PARI) b(n, k) = (-1)^n*(n!/k!)^2 *binomial(n-1, k-1);
t(n, k) = b(n, k) + b(n, n-k+1);
for(n=1, 10, for(k=1, n, print1(t(n, k) - t(n, 1) + 1, ", "))) \\ G. C. Greubel, May 20 2019
(Magma)
b:= func< n, k | (-1)^n*(Factorial(n)/Factorial(k))^2*Binomial(n-1, k-1) >;
[[b(n, k) +b(n, n-k+1) -b(n, 1) -b(n, n) +1: k in [1..n]]: n in [1..10]]; // G. C. Greubel, May 20 2019
(Sage)
def b(n, k): return (-1)^n*factorial(n-k)^2*binomial(n, k)^2*binomial(n-1, k-1)
def t(n, k): return b(n, k) + b(n, n-k+1)
[[t(n, k) - t(n, 1) + 1 for k in (1..n)] for n in (1..10)] # G. C. Greubel, May 20 2019
CROSSREFS
Cf. A008297.
Sequence in context: A156888 A173890 A159767 * A330199 A336810 A178473
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Apr 05 2010
EXTENSIONS
Edited by G. C. Greubel, May 20 2019
STATUS
approved

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Last modified July 24 22:37 EDT 2024. Contains 374585 sequences. (Running on oeis4.)