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A105558
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Central terms in even-indexed rows of triangle A105556 and thus equals the n-th row sum of the n-th matrix power of the Narayana triangle A001263.
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1
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1, 2, 12, 148, 3105, 99156, 4481449, 272312216, 21414443481, 2116193061340, 256712977920256, 37506637787774112, 6496315164318118165, 1316230822119433518312, 308426950979497974254310
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Each term a(n) is divisible by (n+1) for all n>=0.
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FORMULA
| Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Jan 31 2009: (Start)
a(n) = (n+1)*A155926(n) for n>=0.
G.f.: A(x) = d/dx x*F(x) where F(x) = B(x*F(x)) and F(x) = Sum_{n>=0} A155926(n)*x^n/[n!*(n+1)!/2^n] with B(x) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n] and A(x) = Sum_{n>=0} a(n)*x^n/[n!*(n+1)!/2^n]. (End)
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EXAMPLE
| Terms a(n) divided by (n+1) begin:
1,1,4,37,621,16526,640207,34039027,2379382609,211619306134,...
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Jan 31 2009: (Start)
G.f.: A(x) = 1 + 2*x + 12*x^2/3 + 148*x^3/18 + 3105*x^4/180 +...+ a(n)*x^n/[n!*(n+1)!/2^n] +...
G.f.: A(x) = d/dx x*F(x) where F(x) = B(x*F(x)) and:
F(x) = 1 + x + 4*x^2/3 + 37*x^3/18 + 621*x^4/180 + 16526*x^5/2700 +...+ A155926(n)*x^n/[n!*(n+1)!/2^n] +...
B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 +...+ x^n/[n!*(n+1)!/2^n] +... (End)
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PROG
| (PARI) {a(n)=local(N=matrix(n+1, n+1, m, j, if(m>=j, binomial(m-1, j-1)*binomial(m, j-1)/j))); sum(j=0, n, (N^n)[n+1, j+1])}
(PARI) {a(n)=local(F=sum(k=0, n, x^k/(k!*(k+1)!/2^k))+x*O(x^n)); polcoeff(deriv(serreverse(x/F)), n)*n!*(n+1)!/2^n} [From Paul D. Hanna (pauldhanna(AT)juno.com), Jan 31 2009]
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CROSSREFS
| Cf. A001263, A105556.
Cf. A155926. [From Paul D. Hanna (pauldhanna(AT)juno.com), Jan 31 2009]
Sequence in context: A010790 A086928 A001927 * A126777 A126345 A000795
Adjacent sequences: A105555 A105556 A105557 * A105559 A105560 A105561
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Apr 14 2005
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