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A110501
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Unsigned Genocchi numbers (of first kind) of even index.
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8
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1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619, 1109652905, 51943281731, 2905151042481, 191329672483963, 14655626154768697, 1291885088448017715, 129848163681107301953, 14761446733784164001387, 1884515541728818675112649, 268463531464165471482681379
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| The Genocchi numbers satisfy Seidel's recurrence: for n>1, 0 = sum{j=0..[n/2], (-1)^j*C(n,2*j)*a(n-j)}. - R. Stephan, Apr 17 2004
The (n+1)st Genocchi number is the number of Dumont permutations of the first kind on 2n letters. In a Dumont permutation of first kind, each even integer must be followed by a smaller integer and each odd integer is either followed by a larger integer or is the last element. - R. Stephan, Apr 26 2004
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REFERENCES
| R. C. Archibald, Review of Terrill-Terrill paper, Math. Comp., 1 (1945), pp. 385-386.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
L. Euler, Institutionum Calculi Differentialis, volume 2 (1755), para. 181.
A. Genocchi, Intorno all'espressione generale de'numeri Bernulliani, Ann. Sci. Mat. Fis., 3 (1852), 395-405.
Florent Hivert and Olivier Mallet, Combinatorics of k-shapes and Genocchi numbers, in FPSAC 2011, Reykjav´k, Iceland DMTCS proc. AO, 2011, 493-504; http://www-igm.univ-mlv.fr/~fpsac/FPSAC11/SITE2011/proceedings/dmAO0144.pdf.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; p. 74 see Problem 5.8.
H. M. Terrill and E. M. Terrill, Tables of numbers related to the tangent coefficients, J. Franklin Inst., 239 (1945), 64-67.
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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 1..100
Guo-Niu Han, Enumeration of Standard Puzzles
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FORMULA
| a(n) = 2*(-1)^n*(1-4^n)*B_{2*n} (B = Bernoulli numbers).
A002105(n) = 2^(n-1)/n * a(n). - D. E. Knuth, Jan 16 2007
E.g.f.: x*tan(x/2) = Sum_{k > 0} a(k)*x^(2*k)/(2*k)!.
O.g.f.: Sum_{n>=0} n!^2 * x^(n+1) / Product_{k=1..n} (1 + k^2*x). [From Paul D. Hanna, Jul 21 2011]
a(n) = Sum_{k=0..2*n} (-1)^(n-k+1)*Stirling2(2*n, k)*A059371(k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 07 2004
O.g.f.: A(x) = x/(1-x/(1-2*x/(1-4*x/(1-6*x/(1-9*x/(1-12*x/(... -[(n+1)/2]*[(n+2)/2]*x/(1- ...)))))))) (continued fraction). - Paul D. Hanna, Jan 16 2006
a(n) = Pi^(-2*n)*integral(log(t/(1-t))^(2*n)-log(1-1/t)^(2*n) dt,t=0,1) - [Gerry Martens, May 25 2011]
a(n) = the upper left term of M^(n-1); M is an infinite square production matrix with M[i,j] = C(i+1,j-1), i.e. Pascal’s triangle without the first two rows and right border, see the examples and Maple program. [Gary W. Adamson, Jul 19 2011]
E.g.f.: x/sin(x)-x*cot(x)=x^2/G(0); G(k)=4k+2-x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011
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EXAMPLE
| E.g.f.: x*tan(x/2) = x^2/2! + x^4/4! + 3*x^6/6! + 17*x^8/8! + 155*x^10/10! +...
O.g.f.: A(x) = x + x^2 + 3*x^3 + 17*x^4 + 155*x^5 + 2073*x^6 +...
where A(x) = x + x^2/(1+x) + 2!^2*x^3/((1+x)*(1+4*x)) + 3!^2*x^4/((1+x)*(1+4*x)*(1+9*x)) + 4!^2*x^5/((1+x)*(1+4*x)*(1+9*x)*(1+16*x)) +... [From Paul D. Hanna, Jul 21 2011]
From Gary W. Adamson, Jul 19 2011: (Start)
The first few rows of production matrix M are:
1, 2, 0, 0, 0, 0,...
1, 3, 3, 0, 0, 0,...
1, 4, 6, 4, 0, 0,...
1, 5, 10, 10, 5, 0,...
1, 6, 15, 20, 15, 6,... (End)
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MAPLE
| nmax:=17: M := Matrix(1..nmax+1, 1..nmax+1): for i from 1 to nmax do for j from 1 to i+1 do M[i, j] := binomial(i+1, j-1) od: od: for n from 1 to nmax do B := M^(n-1): a(n) := B[1, 1] od: seq(a(n), n=1..nmax); [Johannes W. Meijer, Jul 20 2011]
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PROG
| (PARI) a(n)=if(n<1, 0, 2*(-1)^n*(1-4^n)*bernfrac(2*n))
(PARI) {a(n)=if(n<1, 0, (2*n)!*polcoeff( x*tan(x/2+x*O(x^(2*n))), 2*n))}
(PARI) {a(n)=polcoeff(sum(m=0, n, m!^2*x^(m+1)/prod(k=1, m, 1+k^2*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 21 2011 */
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CROSSREFS
| Cf. A036968(2*n) = A001469(n) = (-1)^n*a(n).
Sequence in context: A135751 A168441 A001469 * A066211 A163884 A175607
Adjacent sequences: A110498 A110499 A110500 * A110502 A110503 A110504
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Jul 23 2005
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