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 A154987 Triangle read by rows: T(n,k) = t(n,k) + t(n,n-k), where t(n,k) = 2*(n!/k!)*(2*(n + k) - 1). 1
 -2, 4, 4, 13, 20, 13, 41, 69, 69, 41, 183, 268, 264, 268, 183, 1099, 1405, 1080, 1080, 1405, 1099, 7943, 9486, 5970, 4080, 5970, 9486, 7943, 65547, 75775, 43806, 20370, 20370, 43806, 75775, 65547, 604831, 685672, 384552, 149520, 77280, 149520, 384552, 685672, 604831 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 LINKS G. C. Greubel, Rows n = 0..100 of the triangle, flattened FORMULA T(n,k) = t(n,k) + t(n,n-k), where t(n,k) = 2*n!*Gamma(n + k + 1/2)/(k!*Gamma(n + k - 1/2)). T(n,k) = t(n,k) + t(n,n-k), where t(n,k) = 2*(n+k-1/2)*(n!/k!). - Yu-Sheng Chang, Apr 13 2020 From G. C. Greubel, May 28 2020: (Start) T(n,k) = binomial(n,k)*( (2*n+2*k-1)*(n-k)! + (4*n-2*k-1)*k! ). T(n,n-k) = T(n,k), for k >= 0. Sum_{k=0..n} T(n,k) = 2*( 2*n-1 + (2*n+1)*n!*e_{n-1}(1) ), where e_{n}(x) is the finite exponential function = Sum_{k=0..n} x^k/k!. Sum_{k=0..n} T(n,k) = 2*( 2*n-1 + (2*n+1)*A007526(n) ). T(n,0) = A175925(n-1) + 2*n. T(n,1) = A007680(n) + A001107(n). (End) EXAMPLE -2;       4,     4;      13,    20,    13;      41,    69,    69,    41;     183,   268,   264,   268,   183;    1099,  1405,  1080,  1080,  1405,  1099;    7943,  9486,  5970,  4080,  5970,  9486,  7943;   65547, 75775, 43806, 20370, 20370, 43806, 75775, 65547;   ... MAPLE t:= proc(n, k) option remember; ## simplified t; 2*(n+k-1/2)*(n!/k!); end proc: A154987:= proc(n, k) ## n >= 0 and k = 0 .. n t(n, k) + t(n, n-k) end proc: # Yu-Sheng Chang, Apr 13 2020 MATHEMATICA (* First program *) t[n_, k_]:= 2*n!*Gamma[n+k+1/2]/(k!*Gamma[n+k-1/2]); T[n_, k_]:= t[n, k] + t[n, n-k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Second Program *) T[n_, k_]:= Binomial[n, k]*((n-k)!*(2*n+2*k-1) + k!*(4*n-2*k-1)); Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, May 28 2020 *) PROG (Sage) def T(n, k): return binomial(n, k)*(factorial(n-k)*(2*n+2*k-1) + factorial(k)*(4*n-2*k-1)) [[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 28 2020 CROSSREFS Sequence in context: A086915 A059927 A290437 * A089419 A263023 A193848 Adjacent sequences:  A154984 A154985 A154986 * A154988 A154989 A154990 KEYWORD sign,tabl AUTHOR Roger L. Bagula, Jan 18 2009 EXTENSIONS Partially edited by Andrew Howroyd, Mar 26 2020 Additionally edited by G. C. Greubel, May 28 2020 STATUS approved

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Last modified October 21 05:38 EDT 2020. Contains 337911 sequences. (Running on oeis4.)