The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 G. C. Greubel, Rows n = 1..100 of the triangle, flattened 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 Cf. A000262, A001263, A008297, A105278, A169653. Sequence in context: A016519 A113716 A220652 * A357744 A088158 A136449 Adjacent sequences: A169651 A169652 A169653 * A169655 A169656 A169657 KEYWORD sign,tabl,easy,less AUTHOR Roger L. Bagula, Apr 05 2010 EXTENSIONS Edited by G. C. Greubel, Feb 23 2021 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

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