A174480
Rectangular array of coefficients in successive iterations of x*exp(x), as read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 15, 23, 1, 1, 5, 28, 102, 104, 1, 1, 6, 45, 274, 861, 537, 1, 1, 7, 66, 575, 3400, 8598, 3100, 1, 1, 8, 91, 1041, 9425, 50734, 98547, 19693, 1, 1, 9, 120, 1708, 21216, 187455, 880312, 1270160, 136064, 1, 1, 10, 153, 2612, 41629
Offset: 1
Form an array of coefficients in the iterations of x*exp(x), which begin:
n=1: [1, 1, 1/2!, 1/3!, 1/4!, 1/5!, 1/6!, ...];
n=2: [1, 2, 6/2!, 23/3!, 104/4!, 537/5!, 3100/6!, ...];
n=3: [1, 3, 15/2!, 102/3!, 861/4!, 8598/5!, 98547/6!, ...];
n=4: [1, 4, 28/2!, 274/3!, 3400/4!, 50734/5!, 880312/6!, ...];
n=5: [1, 5, 45/2!, 575/3!, 9425/4!, 187455/5!, 4367245/6!, ...];
n=6: [1, 6, 66/2!, 1041/3!, 21216/4!, 527631/5!, 15441636/6!, ...];
n=7: [1, 7, 91/2!, 1708/3!, 41629/4!, 1242892/5!, 43806175/6!, ...];
n=8: [1, 8, 120/2!, 2612/3!, 74096/4!, 2582028/5!, 106459312/6!, ...];
n=9: [1, 9, 153/2!, 3789/3!, 122625/4!, 4885389/5!, 230689017/6!, ...];
n=10:[1, 10, 190/2!, 5275/3!, 191800/4!, 8599285/5!, 457584940/6!,...];
...
This array begins with the above unreduced numerators for n >= 1, k >= 1.
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{T(n, k)=local(F=x, xEx=x*exp(x+x*O(x^(k+1)))); for(i=1,n,F=subst(F, x, xEx));(k-1)!*polcoeff(F, k)}
A174494
a(n) = coefficient of x^n/(n-1)! in the 4-fold iteration of x*exp(x).
Original entry on oeis.org
1, 4, 28, 274, 3400, 50734, 880312, 17357736, 382463824, 9298086490, 246914949376, 7104423326356, 220000621675912, 7290852811359654, 257332393857067720, 9632914084301343304, 381050245422453157408
Offset: 1
E.g.f.: x + 4*x^2 + 28*x^3/2! + 274*x^4/3! + 3400*x^5/4! +...
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{a(n)=local(F=x, xEx=x*exp(x+x*O(x^n)));for(i=1,4,F=subst(F, x, xEx));(n-1)!*polcoeff(F, n)}
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{a(n)=sum(k=0,n-1,binomial(n-1,k)*sum(j=0,n-1-k,binomial(n-1-k,j)*(k+1)^j*sum(i=0,n-1-k-j,binomial(n-1-k-j,i)*(k+1+j)^i*(k+1+j+i)^(n-1-k-j-i))))}
A174495
a(n) = coefficient of x^n/(n-1)! in the 5-fold iteration of x*exp(x).
Original entry on oeis.org
1, 5, 45, 575, 9425, 187455, 4367245, 116322645, 3479863345, 115353325835, 4192244804645, 165607074622665, 7060695856372105, 322973775761169135, 15770136907303728205, 818373668098974428885, 44963322539225628107105
Offset: 1
E.g.f.: x + 5*x^2 + 45*x^3/2! + 575*x^4/3! + 9425*x^5/4! +...
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{a(n)=local(F=x, xEx=x*exp(x+x*O(x^n))); for(i=1,5,F=subst(F, x, xEx));(n-1)!*polcoeff(F, n)}
A174496
a(n) = coefficient of x^n/(n-1)! in the 6-fold iteration of x*exp(x).
Original entry on oeis.org
1, 6, 66, 1041, 21216, 527631, 15441636, 518651881, 19630068656, 825581830491, 38159948599956, 1921319136589221, 104603652465885096, 6120324106269585751, 382829011514506048556, 25484466375276284094561
Offset: 1
E.g.f.: x + 6*x^2 + 66*x^3/2! + 1041*x^4/3! + 21216*x^5/4! +...
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{a(n)=local(F=x, xEx=x*exp(x+x*O(x^n))); for(i=1,6,F=subst(F, x, xEx));(n-1)!*polcoeff(F, n)}
Showing 1-4 of 4 results.
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