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A169654 Triangle T(n, k) = A169643(n, k) - A169653(n, 1) + 1, read by rows. 1
1, 1, 1, 1, -4, 1, 1, 24, 24, 1, 1, -138, -118, -138, 1, 1, 1110, 780, 780, 1110, 1, 1, -10120, -8188, -3358, -8188, -10120, 1, 1, 100856, 101976, 30240, 30240, 101976, 100856, 1, 1, -1088710, -1332574, -512062, -60478, -512062, -1332574, -1088710, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
LINKS
FORMULA
T(n, k) = t(n, k) + t(n, n-k+1) - t(n, 1) - t(n, n) + 1, where t(n, k) = (-1)^n*(n!/k!)*binomial(n-1, k-1).
T(n, k) = A008297(n,k) + A008297(n,n-k+1) - (A008297(n,1) + A008297(n,n)) + 1.
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = A169653(n, k) - A169653(n, 1) + 1
T(n, k) = A169653(n, k) - (-1)^n * (n! + 1) + 1.
T(n, k) = (-1)^n * (A105278(n, k) + A105278(n, n-k+1) - (n! + 1) + (-1)^n).
Sum_{k=1..n} T(n, k) = (-1)^n *(2 * A000262(n) - n*(n! + 1) + (-1)^n * n). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, -4, 1;
1, 24, 24, 1;
1, -138, -118, -138, 1;
1, 1110, 780, 780, 1110, 1;
1, -10120, -8188, -3358, -8188, -10120, 1;
1, 100856, 101976, 30240, 30240, 101976, 100856, 1;
1, -1088710, -1332574, -512062, -60478, -512062, -1332574, -1088710, 1;
1, 12700890, 18147240, 9132480, 816480, 816480, 9132480, 18147240, 12700890, 1;
MATHEMATICA
t[n_, m_] = (-1)^n*(n!/m!)*Binomial[n-1, m-1];
T[n_, m_] = t[n, m] + t[n, n-m+1] - (-1)^n*(n! + 1) + 1;
Table[T[n, k], {n, 12}], {k, n}]//Flatten (* modified by G. C. Greubel, Feb 23 2021 *)
PROG
(Sage)
def A001263(n, k): return binomial(n-1, k-1)*binomial(n, k-1)/k
def A169653(n, k): return (-1)^n*A001263(n, k)*(factorial(k) + factorial(n-k+1))
def A169654(n, k): return A169653(n, k) - A169653(n, 1) + 1
flatten([[A169654(n, k) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Feb 23 2021
(Magma)
A001263:= func< n, k | Binomial(n-1, k-1)*Binomial(n, k-1)/k >;
A169653:= func< n, k | (-1)^n*A001263(n, k)*(Factorial(k) + Factorial(n-k+1)) >;
A169654:= func< n, k | A169653(n, k) - A169653(n, 1) + 1 >;
[A169654(n, k): k in [1..n], n in [1..10]]; // G. C. Greubel, Feb 23 2021
CROSSREFS
Sequence in context: A016519 A113716 A220652 * A357744 A088158 A136449
KEYWORD
sign,tabl,easy,less
AUTHOR
Roger L. Bagula, Apr 05 2010
EXTENSIONS
Edited by G. C. Greubel, Feb 23 2021
STATUS
approved

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Last modified June 18 04:26 EDT 2024. Contains 373468 sequences. (Running on oeis4.)