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 A169653 Triangle T(n,k) = A008297(n,k) + A008297(n,n-k+1), read by rows. 2
 -2, 3, 3, -7, -12, -7, 25, 48, 48, 25, -121, -260, -240, -260, -121, 721, 1830, 1500, 1500, 1830, 721, -5041, -15162, -13230, -8400, -13230, -15162, -5041, 40321, 141176, 142296, 70560, 70560, 142296, 141176, 40321, -362881, -1451592, -1695456, -874944, -423360, -874944, -1695456, -1451592, -362881 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS G. C. Greubel, Rows n = 1..100 of the triangle, flattened FORMULA T(n, k) = t(n, k) + t(n, n-k+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). From G. C. Greubel, Feb 23 2021: (Start) T(n, k) = (-1)^n * (A105278(n, k) + A105278(n, n-k+1)). T(n, k) = (-1)^n * ( k! + (n-k+1)! ) * A001263(n, k). Sum_{k=1..n} T(n, k) = 2 * (-1)^n * A000262(n). (End) EXAMPLE Triangle begins as: -2; 3, 3; -7, -12, -7; 25, 48, 48, 25; -121, -260, -240, -260, -121; 721, 1830, 1500, 1500, 1830, 721; -5041, -15162, -13230, -8400, -13230, -15162, -5041; 40321, 141176, 142296, 70560, 70560, 142296, 141176, 40321; MATHEMATICA t[n_, m_] = (-1)^n*(n!/m!)*Binomial[n-1, m-1]; T[n_, m_] = t[n, m] + t[n, n-m+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)) flatten([[A169653(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)) >; [A169653(n, k): k in [1..n], n in [1..10]]; // G. C. Greubel, Feb 23 2021 CROSSREFS Cf. A000262, A001263, A008297, A105278, A169154. Sequence in context: A095978 A370361 A156763 * A129012 A136122 A121875 Adjacent sequences: A169650 A169651 A169652 * A169654 A169655 A169656 KEYWORD sign,tabl AUTHOR Roger L. Bagula, Apr 05 2010 EXTENSIONS Edited by G. C. Greubel, Feb 23 2021 STATUS approved

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Last modified July 18 13:34 EDT 2024. Contains 374378 sequences. (Running on oeis4.)