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Search: id:A108267
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| A108267 |
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Triangle, read by rows, where row n has g.f.: (1-x)^(n+1)*[Sum_{j=0..n} C(n+n*j+j,n*j+j)*x^j]. |
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+0
8
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| 1, 1, 1, 1, 7, 1, 1, 31, 31, 1, 1, 121, 381, 121, 1, 1, 456, 3431, 3431, 456, 1, 1, 1709, 26769, 60691, 26769, 1709, 1, 1, 6427, 193705, 848443, 848443, 193705, 6427, 1, 1, 24301, 1343521, 10350421, 19610233, 10350421, 1343521, 24301, 1, 1, 92368
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Row sums are A000169(n) = (n+1)^n. Column 1 forms A048775(n) = binomial(2*n+1,n+1)-(n+1).
G.f. of row n divided by (1-x)^(n+1) equals g.f. of row n of table A060543. Matrix product of this triangle with Pascal's triangle (A007318) equals A108291.
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FORMULA
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Sum_{k=0..n} T(n, k)*2^k = A108292(n).
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EXAMPLE
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Triangle begins:
1;
1,1;
1,7,1;
1,31,31,1;
1,121,381,121,1;
1,456,3431,3431,456,1;
1,1709,26769,60691,26769,1709,1;
1,6427,193705,848443,848443,193705,6427,1; ...
G.f. of row 3: (1 + 31*x + 31*x^2 + x^3) = (1-x)^4*(1 + 35*x + 165*x^2 + 455*x^3 +... + C(4*j+3,4*j)*x^j +...).
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PROGRAM
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(PARI) T(n, k)=polcoeff((1-x)^(n+1)*sum(j=0, n, binomial(n+n*j+j, n*j+j)*x^j), k)
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CROSSREFS
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Cf. A108267, A000169, A048775.
Cf. A060543, A108291, A108292.
Sequence in context: A142465 A154337 A033933 this_sequence A156916 A166973 A157156
Adjacent sequences: A108264 A108265 A108266 this_sequence A108268 A108269 A108270
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), May 29 2005 and May 31 2005
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