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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; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Triangle is second binomial transform of A008290. - Paul Barry (pbarry(AT)wit.ie), May 25 2006
-exp(-x)*Sum(k=0,n,T[n,k]*x^k) = Integral (x+1)^n*exp(-x) dx = -exp(1)*Gamma(n+1,x+1). [From Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Mar 15 2009]
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FORMULA
| 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
| 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
| Cf. A008290, A008291.
Column 0 is A000522.
Sequence in context: A137650 A171515 A110271 * A103718 A113350 A164678
Adjacent sequences: A073104 A073105 A073106 * A073108 A073109 A073110
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 19 2002
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 23 2004
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