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A091768
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Similar to Bell numbers (A000110).
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4
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1, 2, 6, 22, 92, 426, 2150, 11708, 68282, 423948, 2788230, 19341952, 141003552, 1076787624, 8589843716, 71404154928, 617151121998, 5535236798058, 51426766394244, 494145546973656
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Equals row sums of triangle A163946 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 06 2009]
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REFERENCES
| P. Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, Arxiv preprint arXiv:1107.5490, 2011.
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FORMULA
| G.f. satisfies: (1-x)*A(x-x^2) = 1 + x*A(x). G.f. satisfies: A(x) = C(x) + x*C(x)^2*A(x*C(x)), where C(x) is the Catalan function (A000108). a(n) = A000108(n) + Sum_{k=0..n-1} a(k)*C(2*n-k-1,n-k-1)*(k+2)/(n+1) for n>=0; eigensequence (shift left) of the Catalan triangle A033184. [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 13 2008]
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EXAMPLE
| The Bell numbers can be generated by;
1
1 2
2 3 5
5 7 10 15
where the Bell numbers are the last entry on each line. This last entry is the first entry on the next line and then the entries of the previous line are added, e.g. 7=5+2, 10=7+3, 15=10+5.
This version adds ALL of the entries in the previous column to the new entry.
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1 2
2 4 6
6 10 16 22
where 10=6+2+1+1, 16=10+2+4, 22=16+6
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PROG
| (PARI) { v=vector(20); for (i=1, 20, v[i]=vector(i)); v[1][1]=1; for (i=2, 20, v[i][1]=v[i-1][i-1]; for (j=2, i, v[i][j]=v[i][j-1]+sum(k=j-1, i-1, v[k][j-1]))); for (i=1, 20, print1(", "v[i][i])) }
(PARI) {a(n)=binomial(2*n, n)/(n+1)+sum(k=0, n-1, a(k)*binomial(2*n-k-1, n-k-1)*(k+2)/(n+1))} (PARI) {a(n)=local(A=1+x*O(x^n), C=serreverse(x-x^2+x^2*O(x^n))/x); for(i=0, n, A=C+x*C^2*subst(A, x, x*C)); polcoeff(A, n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 13 2008]
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CROSSREFS
| Close to A074664
Cf. A000110 (Bell Numbers).
Cf. A033184, A000108. [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 13 2008]
A163946 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 06 2009]
Sequence in context: A124294 A124295 A074664 * A185349 A150274 A109317
Adjacent sequences: A091765 A091766 A091767 * A091769 A091770 A091771
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KEYWORD
| nonn
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AUTHOR
| Jon Perry (perry(AT)globalnet.co.uk), Mar 06 2004
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