A220266 G.f.: Sum_{n>=1} (2*(1+x)^n - 1) * ((1+x)^n - 1)^(n-1).
1, 4, 18, 144, 1604, 22944, 400624, 8259680, 196358760, 5287879092, 159094582274, 5288950560768, 192527721428892, 7616404083126180, 325361411700398046, 14926683772801407168, 731947910056020737036, 38204289826040411251632, 2114787166947079113869760
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 4*x + 18*x^2 + 144*x^3 + 1604*x^4 + 22944*x^5 +... where A(x) = (1+2*x) + (1+4*x+2*x^2)*(2*x+x^2) + (1+6*x+6*x^2+2*x^3)*(3*x+3*x^2+x^3)^2 + (1+8*x+12*x^2+8*x^3+2*x^4)*(4*x+6*x^2+4*x^3+x^4)^3 +... Compare the g.f. to the identity: 1 = (1+2*x) - (1+4*x+2*x^2)*(2*x+x^2) + (1+6*x+6*x^2+2*x^3)*(3*x+3*x^2+x^3)^2 - (1+8*x+12*x^2+8*x^3+2*x^4)*(4*x+6*x^2+4*x^3+x^4)^3 +-...
Programs
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PARI
{a(n)=polcoeff(sum(m=1,n+1,(2*(1+x)^m - 1) * ((1+x)^m - 1 +x*O(x^n))^(m-1)),n)} for(n=0,20,print1(a(n),", ")) -
PARI
/* As Row Sums of Triangle A220265: */ {A220265(n,k)=polcoeff((2*(1+x)^n-1)*((1+x)^n-1)^(n-1)/x^(n-1),k)} {a(n)=sum(k=0,n,A220265(n-k+1,k))} for(n=0,20,print1(a(n),", "))
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PARI
{a(n)=polcoeff(1+sum(m=1,n\2+1,2*(2*(1+x)^(2*m) - 1) * ((1+x)^(2*m) - 1 +x*O(x^n))^(2*m-1)),n)} for(n=0,20,print1(a(n),", ")) -
PARI
{a(n)=polcoeff(-1+sum(m=0,n\2,2*(2*(1+x)^(2*m+1) - 1) * ((1+x)^(2*m+1) - 1 +x*O(x^n))^(2*m)),n)} for(n=0,20,print1(a(n),", "))
Comments