Search: id:A036969
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%I A036969
%S A036969 1,1,1,1,5,1,1,21,14,1,1,85,147,30,1,1,341,1408,627,55,1,1,1365,13013,
%T A036969 11440,2002,91,1,1,5461,118482,196053,61490,5278,140,1,1,21845,1071799,
%U A036969 3255330,1733303,251498,12138,204,1,1,87381,9668036,53157079,46587905
%N A036969 Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 <= k <= 
               n.
%C A036969 Or, triangle central factorial numbers T(2n,2k) (in Riordan's notation).
%C A036969 Can be used to calculate the Bernoulli numbers via the formula B_2n = 
               (1/2)*Sum{k= 1..n, (-1)^(k+1)*(k-1)!*k!*T(n,k)/(2*k+1)}. E.g. : n 
               = 1: B_2= (1/2)*1/3 = 1/6. n = 2: B_4 = (1/2)*(1/3 - 2/5) = -1/30. 
               n = 3: B_6 = (1/2)*(1/3 - 2*5/5 + 2*6/7) = 1/42. - DELEHAM Philippe 
               (kolotoko(AT)wanadoo.fr), Nov 13 2003
%D A036969 D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke 
               Math. J., 41 (1974), 305-318.
%D A036969 P. A. MacMahon, Divisors of numbers and their continuations in the theory 
               of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. 
               Papers II, pp. 303-341.
%D A036969 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A036969 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Problem 5.8.
%e A036969 1; 1,1; 1,5,1; 1,21,14,1; 1,85,147,30,1; ...
%p A036969 A036969 := proc(n,k) local j; 2*add(j^(2*n)*(-1)^(k-j)/((k-j)!*(k+j)!),
               j=1..k); end;
%Y A036969 Diagonals are A002450, A002451, A000330 and A060493. Cf. A008955.
%Y A036969 Transpose of A008957. Cf. A008955, A008956, A008957.
%Y A036969 Sequence in context: A144397 A047909 A111577 this_sequence A080249 A157154 
               A022168
%Y A036969 Adjacent sequences: A036966 A036967 A036968 this_sequence A036970 A036971 
               A036972
%K A036969 nonn,easy,nice,tabl
%O A036969 1,5
%A A036969 N. J. A. Sloane (njas(AT)research.att.com).
%E A036969 More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 16 2000

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