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 A081297 Array T(k,n), read by antidiagonals: T(k,n) = ((k+1)^(n+1)-(-k)^(n+1))/(2k+1). 8
 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 5, 1, 1, 1, 13, 13, 11, 1, 1, 1, 21, 25, 55, 21, 1, 1, 1, 31, 41, 181, 133, 43, 1, 1, 1, 43, 61, 461, 481, 463, 85, 1, 1, 1, 57, 85, 991, 1281, 2653, 1261, 171, 1, 1, 1, 73, 113, 1891, 2821, 10501, 8425, 4039, 341, 1, 1, 1, 91, 145, 3305 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS Square array of solutions of a family of recurrences. Rows of the array give solutions to the recurrences a(n)=a(n-1)+k(k-1)a(n-2), a(0)=a(1)=1. Rows include A001045, A015441, A053404, A053428, A053430. Columns include A002061, A001844, A072025. Diagonals include A081298, A081299, A081300, A081301, A081302. Subarray of array in A072024. - Philippe Deléham, Nov 24 2013 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1274 FORMULA T(k, n)=((k+1)^(n+1)-(-k)^(n+1))/(2k+1). Rows of the array have g.f. 1/((1+kx)(1-(k+1)x)). EXAMPLE Rows begin   1, 1,  1,  1,   1,    1, ...   1, 1,  3,  5,  11,   21, ...   1, 1,  7, 13,  55,  133, ...   1, 1, 13, 25, 181,  481, ...   1, 1, 21, 41, 461, 1281, ... MATHEMATICA T[n_, k_]:=((n + 1)^(k + 1) - (-n)^(k + 1)) / (2n + 1); Flatten[Table[T[n - k, k], {n, 0, 10}, {k, 0, n}]] (* Indranil Ghosh, Mar 27 2017 *) PROG (PARI) for(k=0, 10, for(n=0, 9, print1(((k+1)^(n+1)-(-k)^(n+1))/(2*k+1), ", "); ); print(); ) \\ Andrew Howroyd, Mar 26 2017 (Python) def T(n, k): return ((n + 1)**(k + 1) - (-n)**(k + 1)) / (2*n + 1) for n in xrange(0, 11): ....print [T(n - k, k) for k in xrange(0, n + 1)] # Indranil Ghosh, Mar 27 2017 CROSSREFS Cf. A059259, A072024. Sequence in context: A243473 A325969 A325826 * A110180 A005765 A263159 Adjacent sequences:  A081294 A081295 A081296 * A081298 A081299 A081300 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, Mar 17 2003 EXTENSIONS Name clarified by Andrew Howroyd, Mar 27 2017 STATUS approved

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Last modified June 17 22:17 EDT 2019. Contains 324200 sequences. (Running on oeis4.)