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 A350557 Triangle T(n,k) read by rows with T(n,0) = (2*n)! / (2^n * n!) for n >= 0 and T(n,k) = (Sum_{i=k..n} binomial(i-1,k-1) * 2^i * i! / (2*i)!) * (2*n)! / (2^n * n!) for 0 < k <= n. 0
 1, 1, 1, 3, 4, 1, 15, 21, 7, 1, 105, 148, 52, 10, 1, 945, 1333, 472, 96, 13, 1, 10395, 14664, 5197, 1066, 153, 16, 1, 135135, 190633, 67567, 13873, 2009, 223, 19, 1, 2027025, 2859496, 1013512, 208116, 30170, 3380, 306, 22, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Table of n, a(n) for n=0..44. FORMULA T(n,n) = 1. T(n,k) = binomial(n-1,k-1) + (2*n - 1) * T(n-1,k) for 0 < k < n. Conjecture: M(n,k) = (-1)^(n-k) * T(n,k) is matrix inverse of A350512. EXAMPLE Triangle T(n,k) for 0 <= k <= n starts: n\k : 0 1 2 3 4 5 6 7 8 ================================================================= 0 : 1 1 : 1 1 2 : 3 4 1 3 : 15 21 7 1 4 : 105 148 52 10 1 5 : 945 1333 472 96 13 1 6 : 10395 14664 5197 1066 153 16 1 7 : 135135 190633 67567 13873 2009 223 19 1 8 : 2027025 2859496 1013512 208116 30170 3380 306 22 1 etc. MATHEMATICA Flatten[Table[If[k==0, (2n)!/(2^n n!), Sum[Binomial[i-1, k-1]2^i i!/(2i)!, {i, k, n}](2n)!/(2^n n!)], {n, 0, 8}, {k, 0, n}]] (* Stefano Spezia, Jan 06 2022 *) CROSSREFS Cf. A001147 (column 0), A286286 (column 1), A249349 (column 2). Cf. A000007 (alternating row sums). Cf. A350512. Sequence in context: A100326 A303728 A321627 * A028338 A039757 A136228 Adjacent sequences: A350554 A350555 A350556 * A350558 A350559 A350560 KEYWORD nonn,easy,tabl AUTHOR Werner Schulte, Jan 05 2022 STATUS approved

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Last modified December 7 11:26 EST 2023. Contains 367650 sequences. (Running on oeis4.)