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A169654
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Triangle T(n, k) = A169643(n, k) - A169653(n, 1) + 1, read by rows.
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1
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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
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table;
graph;
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listen;
history;
text;
internal format)
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OFFSET
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1,5
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LINKS
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FORMULA
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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) = 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)
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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))
(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)) >;
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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