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Search: id:A073107
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| A073107 |
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Triangle T(n,k) read by rows, where e.g.f. for T(n,k) is exp((1+y)*x)/(1-x). |
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3
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| 1, 2, 1, 5, 4, 1, 16, 15, 6, 1, 65, 64, 30, 8, 1, 326, 325, 160, 50, 10, 1, 1957, 1956, 975, 320, 75, 12, 1, 13700, 13699, 6846, 2275, 560, 105, 14, 1, 109601, 109600, 54796, 18256, 4550, 896, 140, 16, 1, 986410, 986409, 493200, 164388, 41076, 8190, 1344
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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Triangle is second binomial transform of A008290. - Paul Barry (pbarry(AT)wit.ie), May 25 2006
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FORMULA
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O.g.f. for k-th column is 1/k!*Sum_{i>=k} i!*x^i/(1-x)^(i+1). For n>0 T(n, 0) = floor(n!*exp(1)) = A000522(n), T(n, 1) = floor(n!*exp(1)-1) = A007526(n), T(n, 2) = 1/2!*floor(n!*exp(1)-1-n) = A038155(n), T(n, 3) = 1/3!*floor(n!*exp(1)-1-n-n*(n-1)), T(n, 4) = 1/4!*floor(n!*exp(1)-1-n-n*(n-1)-n*(n-1)*(n-2)), ... . Row sums give A010842.
E.g.f. for k-th column is x^k/k!*exp(x)/(1-x).
O.g.f. for k-th row is n!*Sum_{k=0..n} (1+x)^k/k!.
T(n,k)=sum{j=0..n, binomial(j,k)*n!/j!}; - Paul Barry (pbarry(AT)wit.ie), May 25 2006
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EXAMPLE
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exp((1+y)*x)/(1-x) = 1+(2+y)*x+1/2!*(5+4*y+y^2)*x^2+1/3!*(16+15*y+6*y^2+y^3)*x^3+1/4!*(65+64*y+30*y^2+8*y^3+y^4)*x^4+1/5!*(326+325*y+160*y^2+50*y^3+10*y^4+y^5)*x^5+...
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CROSSREFS
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Cf. A008290, A008291.
Column 0 is A000522.
Adjacent sequences: A073104 A073105 A073106 this_sequence A073108 A073109 A073110
Sequence in context: A104259 A137650 A110271 this_sequence A103718 A113350 A061579
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 19 2002
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EXTENSIONS
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More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 23 2004
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