A251182
a(n) = Sum_{k=0..n} binomial(n, k) * (2^k - 1)^k.
Original entry on oeis.org
1, 2, 12, 374, 52056, 28885802, 62696043492, 533314127677214, 17882368106959072176, 2375841795610783881752402, 1255349217681407318492850586812, 2644225793295900889867998778302561734, 22235530372401312606563659670627119777739016, 747102526634678016053120249315815798707672485729402
Offset: 0
G.f.: A(x) = 1 + 2*x + 12*x^2 + 374*x^3 + 52056*x^4 + 28885802*x^5 +...
where we have the identity:
(1) A(x) = 1/(1-x) + (2-1)*x/(1-x)^2 + (2^2-1)^2*x^2/(1-x)^3 + (2^3-1)^3*x^3/(1-x)^4 + (2^4-1)^4*x^4/(1-x)^5 + (2^5-1)^5*x^5/(1-x)^6 +...
(2) A(x) = 1 + 2*x/(1-x + 2*x)^2 + 2^4*x^2/(1-x + 2^2*x)^3 + 2^9*x^3/(1-x + 2^3*x)^4 + 2^16*x^4/(1-x + 2^4*x)^5 + 2^25*x^5/(1-x + 2^5*x)^6 + 2^36*x^6/(1-x + 2^6*x)^7 +...
Illustration of initial terms:
a(0) = 1;
a(1) = 1 + (2-1) = 2;
a(2) = 1 + 2*(2-1) + (2^2-1)^2 = 12;
a(3) = 1 + 3*(2-1) + 3*(2^2-1)^2 + (2^3-1)^3 = 374;
a(4) = 1 + 4*(2-1) + 6*(2^2-1)^2 + 4*(2^3-1)^3 + (2^4-1)^4 = 52056;
a(5) = 1 + 5*(2-1) + 10*(2^2-1)^2 + 10*(2^3-1)^3 + 5*(2^4-1)^4 + (2^5-1)^5 = 28885802; ...
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Table[1 + Sum[Binomial[n,k] * (2^k-1)^k,{k,1,n}],{n,0,15}] (* Vaclav Kotesovec, Jan 25 2015 *)
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{a(n)=sum(k=0, n, binomial(n, k) * (2^k - 1)^k )}
for(n=0, 15, print1(a(n), ", "))
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{a(n)=local(A=1); A=sum(m=0, n, (2^m - 1)^m * x^m / (1-x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
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{a(n)=local(A=1); A=sum(m=0, n, 2^(m^2) * x^m / (1-x + x*2^m +x*O(x^n))^(m+1) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
A251183
a(n) = Sum_{k=0..n} binomial(n,k) * (-1)^(n-k) * (2^k + 1)^k.
Original entry on oeis.org
1, 2, 20, 662, 80744, 38724842, 75185316860, 593940189201662, 19026394370724334544, 2460515237682283759296722, 1280059938187721964276628432100, 2672755638642586172344574379667166822, 22366135140308331991861876017552822964355384, 749476887752252949738787898479359344347670538150842
Offset: 0
G.f.: A(x) = 1 + 2*x + 20*x^2 + 662*x^3 + 80744*x^4 + 38724842*x^5 +...
where we have the identity:
(1) A(x) = 1/(1-x) + (2+1)*x/(1+x)^2 + (2^2+1)^2*x^2/(1+x)^3 + (2^3+1)^3*x^3/(1+x)^4 + (2^4+1)^4*x^4/(1+x)^5 + (2^5+1)^5*x^5/(1+x)^6 +...
(2) A(x) = 1 + 2*x/(1+x - 2*x)^2 + 2^4*x^2/(1+x - 2^2*x)^3 + 2^9*x^3/(1+x - 2^3*x)^4 + 2^16*x^4/(1+x - 2^4*x)^5 + 2^25*x^5/(1+x - 2^5*x)^6 +...
Illustration of initial terms:
a(0) = 1;
a(1) = -1 + (2+1) = 2;
a(2) = 1 - 2*(2+1) + (2^2+1)^2 = 20;
a(3) = -1 + 3*(2+1) - 3*(2^2+1)^2 + (2^3+1)^3 = 662;
a(4) = 1 - 4*(2+1) + 6*(2^2+1)^2 - 4*(2^3+1)^3 + (2^4+1)^4 = 80744;
a(5) = -1 + 5*(2+1) - 10*(2^2+1)^2 + 10*(2^3+1)^3 - 5*(2^4+1)^4 + (2^5+1)^5 = 38724842; ...
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Table[Sum[Binomial[n,k] * (-1)^(n-k) * (2^k + 1)^k,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 25 2015 *)
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{a(n)=sum(k=0, n, binomial(n, k) * (-1)^(n-k) * (2^k + 1)^k )}
for(n=0, 15, print1(a(n), ", "))
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{a(n)=local(A=1); A=sum(m=0, n, (2^m + 1)^m * x^m / (1+x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
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{a(n)=local(A=1); A=sum(m=0, n, 2^(m^2) * x^m / (1+x - 2^m*x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
A251671
a(n) = Sum_{k=0..n} C(n,k) * (2^k + 3^k)^k.
Original entry on oeis.org
1, 6, 180, 43398, 88701816, 1573206748746, 248688444559874580, 356335498302585834118638, 4663871943514788530035646937456, 558720685051192771669885091319459750546, 612058892657175926094223171960469926874935754700
Offset: 0
G.f.: A(x) = 1 + 6*x + 180*x^2 + 43398*x^3 + 88701816*x^4 + 1573206748746*x^5 +...
where A(x) = 1/(1-x) + (2+3)*x/(1-x)^2 + (2^2+3^2)*x^2/(1-x)^3 + (2^3+3^3)^3*x^3/(1-x)^4 +...
ILLUSTRATION OF INITIAL TERMS:
a(0) = 1*(2^0+3^0)^0 = 1;
a(1) = 1*(2^0+3^0)^0 + 1*(2^1+3^1)^1 = 6;
a(2) = 1*(2^0+3^0)^0 + 2*(2^1+3^1)^1 + 1*(2^2+3^2)^2 = 180;
a(3) = 1*(2^0+3^0)^0 + 3*(2^1+3^1)^1 + 3*(2^2+3^2)^2 + 1*(2^3+3^3)^3 = 43398;
a(4) = 1*(2^0+3^0)^0 + 4*(2^1+3^1)^1 + 6*(2^2+3^2)^2 + 4*(2^3+3^3)^3 + 1*(2^4+3^4)^4 = 88701816; ...
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Table[Sum[Binomial[n,k] * (2^k + 3^k)^k,{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Jan 25 2015 *)
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{a(n)=sum(k=0, n, binomial(n, k) * (2^k + 3^k)^k )}
for(n=0, 15, print1(a(n), ", "))
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{a(n)=local(A=1); A=sum(m=0, n, (2^m + 3^m)^m * x^m / (1-x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
Showing 1-3 of 3 results.