cp's OEIS Frontend

A081297 Array T(k,n), read by antidiagonals: T(k,n) = ((k+1)^(n+1)-(-k)^(n+1))/(2k+1).

%I A081297 #29 Mar 27 2021 23:52:00
%S A081297 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,
%T A081297 31,41,181,133,43,1,1,1,43,61,461,481,463,85,1,1,1,57,85,991,1281,
%U A081297 2653,1261,171,1,1,1,73,113,1891,2821,10501,8425,4039,341,1,1,1,91,145,3305
%N A081297 Array T(k,n), read by antidiagonals: T(k,n) = ((k+1)^(n+1)-(-k)^(n+1))/(2k+1).
%C A081297 Square array of solutions of a family of recurrences.
%C A081297 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.
%C A081297 Subarray of array in A072024. - _Philippe Deléham_, Nov 24 2013
%H A081297 Andrew Howroyd, <a href="/A081297/b081297.txt">Table of n, a(n) for n = 0..1274</a>
%F A081297 T(k, n) = ((k+1)^(n+1)-(-k)^(n+1))/(2k+1).
%F A081297 Rows of the array have g.f. 1/((1+kx)(1-(k+1)x)).
%e A081297 Rows begin
%e A081297   1, 1,  1,  1,   1,    1, ...
%e A081297   1, 1,  3,  5,  11,   21, ...
%e A081297   1, 1,  7, 13,  55,  133, ...
%e A081297   1, 1, 13, 25, 181,  481, ...
%e A081297   1, 1, 21, 41, 461, 1281, ...
%t A081297 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 *)
%o A081297 (PARI)
%o A081297 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
%o A081297 (Python)
%o A081297 def T(n, k): return ((n + 1)**(k + 1) - (-n)**(k + 1)) // (2*n + 1)
%o A081297 for n in range(11):
%o A081297     print([T(n - k, k) for k in range(n + 1)]) # _Indranil Ghosh_, Mar 27 2017
%Y A081297 Cf. A059259, A072024.
%Y A081297 Rows include A001045, A015441, A053404, A053428, A053430.
%Y A081297 Columns include A002061, A001844, A072025.
%Y A081297 Diagonals include A081298, A081299, A081300, A081301, A081302.
%K A081297 easy,nonn,tabl
%O A081297 0,9
%A A081297 _Paul Barry_, Mar 17 2003
%E A081297 Name clarified by _Andrew Howroyd_, Mar 27 2017