A222053
O.g.f.: Sum_{n>=0} (n^3*x)^n/(1-n^3*x)^n * exp(-n^3*x/(1-n^3*x)) / n!.
Original entry on oeis.org
1, 1, 32, 3536, 877221, 394506859, 284110844070, 302350295364613, 449340338669205876, 894210483750815778132, 2306748823711254973903838, 7516588630649080782251419791, 30292392269310179039574629318038, 148358895760995636729844370111255773
Offset: 0
O.g.f.: A(x) = 1 + x + 32*x^2 + 3536*x^3 + 877221*x^4 + 394506859*x^5 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^6*x^2/(1-2^3*x)^2*exp(-2^3*x/(1-2^3*x))/2! + 3^9*x^3/(1-3^3*x)^3*exp(-3^3*x/(1-3^3*x))/3! + 4^12*x^4/(1-4^3*x)^4*exp(-4^3*x/(1-4^3*x))/4! +...
simplifies to a power series in x with integer coefficients.
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{a(n)=polcoeff(sum(k=0, n+1, (k^3*x)^k/(1-k^3*x)^k*exp(-k^3*x/(1-k^3*x+x*O(x^n)))/k!), n)}
for(n=0, 25, print1(a(n), ", "))
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{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1) * Stirling2(3*n, k)))}
for(n=0, 25, print1(a(n), ", "))
A222054
O.g.f.: Sum_{n>=0} (n^4*x)^n/(1-n^4*x)^n * exp(-n^4*x/(1-n^4*x)) / n!.
Original entry on oeis.org
1, 1, 128, 90621, 193322261, 933620289929, 8632521193856886, 136885314823146617517, 3443427946573913689696192, 129667338445150206244162849650, 6988095504452769015520539806767120, 520011535068804196524689647521015780176
Offset: 0
O.g.f.: A(x) = 1 + x + 128*x^2 + 90621*x^3 + 193322261*x^4 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^8*x^2/(1-2^4*x)^2*exp(-2^4*x/(1-2^4*x))/2! + 3^12*x^3/(1-3^4*x)^3*exp(-3^4*x/(1-3^4*x))/3! + 4^16*x^4/(1-4^4*x)^4*exp(-4^4*x/(1-4^4*x))/4! +...
simplifies to a power series in x with integer coefficients.
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{a(n)=polcoeff(sum(k=0, n+1, (k^4*x)^k/(1-k^4*x)^k*exp(-k^4*x/(1-k^4*x+x*O(x^n)))/k!), n)}
for(n=0, 20, print1(a(n), ", "))
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{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1) * Stirling2(4*n, k)))}
for(n=0, 20, print1(a(n), ", "))
A243942
O.g.f.: Sum_{n>=0} n^(2*n) * x^n / (1 - n*x)^n * exp( -n^2*x / (1 - n*x) ) / n!.
Original entry on oeis.org
1, 1, 8, 121, 2698, 79654, 2929238, 129004633, 6619919386, 387904397222, 25555935470016, 1869945551975658, 150459006927310348, 13203459856456213172, 1254972882696473807298, 128439184335788533011489, 14082139161229781077548346, 1646731810035799151750487814
Offset: 0
O.g.f.: A(x) = 1 + x + 8*x^2 + 121*x^3 + 2698*x^4 + 79654*x^5 + 2929238*x^6 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^4*x^2/(1-2*x)^2*exp(-4*x/(1-2*x))/2! + 3^6*x^3/(1-3*x)^3*exp(-9*x/(1-3*x))/3! + 4^8*x^4/(1-4*x)^4*exp(-16*x/(1-4*x))/4! + 5^10*x^5/(1-5*x)^5*exp(-25*x/(1-5*x))/5! +...
simplifies to a power series in x with integer coefficients.
Illustrate the terms by:
a(1) = 1*1 = 1;
a(2) = 1*1 + 1*7 = 8;
a(3) = 1*1 + 2*15 + 1*90 = 121;
a(4) = 1*1 + 3*31 + 3*301 + 1*1701 = 2698;
a(5) = 1*1 + 4*63 + 6*966 + 4*7770 + 1*42525 = 79654; ...
where Stirling2(n+k,k) forms a rectangular table as follows:
1, 1, 1, 1, 1, 1, 1, 1, ...;
0, 1, 3, 6, 10, 15, 21, 28, ...;
0, 1, 7, 25, 65, 140, 266, 462, ...;
0, 1, 15, 90, 350, 1050, 2646, 5880, ...;
0, 1, 31, 301, 1701, 6951, 22827, 63987, ...;
0, 1, 63, 966, 7770, 42525, 179487, 627396, ...; ...
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Flatten[{1,Table[Sum[Binomial[n-1,k-1] * StirlingS2[n+k,k],{k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Aug 11 2014 *)
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{a(n)=polcoeff(sum(k=0, n+1, (k^2*x)^k/(1-k*x)^k*exp(-k^2*x/(1-k*x+x*O(x^n)))/k!), n)}
for(n=0, 25, print1(a(n), ", "))
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{Stirling2(n, k) = sum(j=0, k, (-1)^(k+j) * binomial(k, j) * j^n) / k!}
{a(n)=if(n==0, 1, sum(k=1, n, Stirling2(n+k, k) * binomial(n-1, k-1)))}
for(n=0, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.