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%I #37 Feb 22 2022 09:58:01
%S 0,1,1,2,3,2,3,5,5,3,4,7,8,7,4,5,9,11,11,9,5,6,11,14,15,14,11,6,7,13,
%T 17,19,19,17,13,7,8,15,20,23,24,23,20,15,8,9,17,23,27,29,29,27,23,17,
%U 9,10,19,26,31,34,35,34,31,26,19,10,11,21,29,35,39,41
%N In a triangle of numbers (such as that in A059032, A059033, A059034) how many entries lie above position (n,k)? Answer: T(n,k) = (n+1)*(k+1)-1 (n >= 0, k >= 0).
%F T(n, k) = max(T(n-1, k-1), T(n-1, k)) + min(k, n-k+1). - _Jon Perry_, Aug 05 2004
%F E.g.f.: exp(x+y)(x+y+xy) (as a square array read by antidiagonals). - _Paul Barry_, Sep 24 2004
%F From _Michael Somos_, Jul 28 2015: (Start)
%F Row sums = Sum_{k=0..n} T(n-k, k) = A005581(n+1).
%F T(n, k) = T(k, n) = T(-2-n, -2-k) for all n, k in Z.
%F Sum_{n, k >= 0} x^T(n, k) = f(x) / x where f() is the g.f. for A000005. (End)
%e As an infinite triangular array:
%e 0
%e 1 1
%e 2 3 2
%e 3 5 5 3
%e 4 7 8 7 4
%e 5 9 11 11 9 5
%e As an infinite square array (matrix):
%e 0 1 2 3 4 5
%e 1 3 5 7 9 11
%e 2 5 8 11 14 17
%e 3 7 11 15 19 23
%e 4 9 14 19 24 29
%e 5 11 17 23 29 35
%o (PARI) {T(n, k) = n + k + n*k}; /* _Michael Somos_, Jul 28 2015 */
%Y T(n, k) = A003991(n, k) - 1.
%Y Cf. A000005, A005581.
%K nonn,tabl,easy
%O 0,4
%A _N. J. A. Sloane_, Feb 13 2001
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