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A138247
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E.g.f.: A(x) = Sum_{n>=0} exp((2^n+3^n)*x) * (2^n+3^n)^n * x^n/n!.
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5
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1, 7, 223, 49849, 94705663, 1616229320497, 251286598125520183, 357716675257916544062689, 4670472774542449929397808845183, 559006854195449142958954163012808059617
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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FORMULA
| a(n) = Sum_{k=0..n} C(n,k)*(2^k + 3^k)^n.
a(n) = Sum_{k=0..n} C(n,k)*(1 + 2^(n-k)*3^k)^n.
a(n) = Sum_{k=0..n} C(n,k)*A007689(k)^n.
a(n) = Sum_{k=0..n} C(n,k)*A094617(n,k)^n.
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EXAMPLE
| a(1) = 2 + 5 = 3 + 4 = 7 ;
a(2) = 2^2 + 2*5^2 + 13^2 = 5^2 + 2*7^2 + 10^2 = 223 ;
a(3) = 2^3 + 3*5^3 + 3*13^3 + 35^3 = 9^3 + 3*13^3 + 3*19^3 + 28^3 = 49849.
When p=2, q=3, this sequence illustrates the following identity:
Sum_{k=0..n} C(n,k)*(p^k + q^k)^n =
Sum_{k=0..n} C(n,k)*(1 + p^(n-k)*q^k)^n
which is a special case of the more general binomial identity:
Sum_{k=0..n} C(n,k)*(s*p^k + t*q^k)^(n-k) * (u*p^k + v*q^k)^k =
Sum_{k=0..n} C(n,k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k.
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PROG
| (PARI) {a(n)=local(p=2, q=3, s=1, t=1, u=1, v=1);
sum(k=0, n, binomial(n, k)*(s*p^k + t*q^k)^(n-k)*(u*p^k + v*q^k)^k)}
/* right side of the general binomial identity: */
{a(n)=local(p=2, q=3, s=1, t=1, u=1, v=1);
sum(k=0, n, binomial(n, k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k)}
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CROSSREFS
| Cf. A007689, A094617.
Sequence in context: A140018 A202655 A009488 * A015506 A193503 A142606
Adjacent sequences: A138244 A138245 A138246 * A138248 A138249 A138250
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Mar 09 2008, revised Mar 11 2008
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