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Search: id:A036969
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| A036969 |
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Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 <= k <= n. |
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6
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| 1, 1, 1, 1, 5, 1, 1, 21, 14, 1, 1, 85, 147, 30, 1, 1, 341, 1408, 627, 55, 1, 1, 1365, 13013, 11440, 2002, 91, 1, 1, 5461, 118482, 196053, 61490, 5278, 140, 1, 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1, 1, 87381, 9668036, 53157079, 46587905
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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Or, triangle central factorial numbers T(2n,2k) (in Riordan's notation).
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
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REFERENCES
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D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
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.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.
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EXAMPLE
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1; 1,1; 1,5,1; 1,21,14,1; 1,85,147,30,1; ...
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MAPLE
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A036969 := proc(n, k) local j; 2*add(j^(2*n)*(-1)^(k-j)/((k-j)!*(k+j)!), j=1..k); end;
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CROSSREFS
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Diagonals are A002450, A002451, A000330 and A060493. Cf. A008955.
Transpose of A008957. Cf. A008955, A008956, A008957.
Sequence in context: A144397 A047909 A111577 this_sequence A080249 A157154 A022168
Adjacent sequences: A036966 A036967 A036968 this_sequence A036970 A036971 A036972
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KEYWORD
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nonn,easy,nice,tabl
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 16 2000
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