A177779
E.g.f.: A(x) = Sum_{n>=0} (1/n!)*Product_{k=0..n-1} L(3^k*x), where L(x) is the e.g.f. of A177780.
Original entry on oeis.org
1, 1, 7, 159, 10065, 1769889, 892392183, 1321983917631, 5822841033057825, 76645599313018616001, 3021493143896197748386407, 357064253282406274455859700319, 126544129732367263008235662373092465
Offset: 0
E.g.f.: A(x) = 1 + x + 7*x^2/2! + 159*x^3/3! + 10065*x^4/4! +...
Then e.g.f. A(x) is given by:
A(x) = 1 + L(x) + L(x)L(3x)/2! + L(x)L(3x)L(9x)/3! + L(x)L(3x)L(9x)L(27x)/4! +...
where L(x) is the e.g.f. of A177780:
. L(x) = x + 4*x^2/2! + 60*x^3/3! + 2496*x^4/4! + 276240*x^5/5! +...
. L(x) = x*d/dx log( Sum_{n>=0} 3^(n(n-1)/2)*x^n/n! )
and satisfies:
. L(x)/x = 1 + 2*L(x) + 2^2*L(x)L(3x)/2! + 2^3*L(x)L(3x)L(9x)/3! + 2^4*L(x)L(3x)L(9x)L(27x)/4! +...
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{a(n,q=3)=local(Lq=x+x^2,A); for(i=1,n,Lq=x*sum(m=0,n,(q-1)^m/m!*prod(k=0,m-1,subst(Lq,x,q^k*x+x*O(x^n))))); A=sum(m=0,n,1/m!*prod(k=0,m-1,subst(Lq,x,q^k*x+x*O(x^n))));n!*polcoeff(A,n)}
A177781
E.g.f. satisfies: L(x) = x*Sum_{n>=0} 3^n/n!*Product_{k=0..n-1} L(4^k*x).
Original entry on oeis.org
1, 6, 162, 15336, 5135400, 6403850928, 30733361357328, 576178771105452672, 42495458789243292762240, 12378928091101498820594407680, 14278666564505879853034906179788544
Offset: 1
E.g.f.: L(x) = x + 6*x^2/2! + 162*x^3/3! + 15336*x^4/4! + 5135400*x^5/5! + ... + n*A003027(n)*x^n/n! + ...
Given the related expansions:
. E(x) = 1 + x + 4*x^2/2! +64*x^3/3! +4096*x^4/4! +1048576*x^5/5! + ...
. log(E(x)) = x + 3*x^2/2! +54*x^3/3! +3834*x^4/4! +1027080*x^5/5! + ... + A003027(n)*x^n/n! + ...
then L(x) satisfies:
. L(x)/x = 1 + 3*L(x) + 3^2*L(x)L(4x)/2! + 3^3*L(x)L(4x)L(16x)/3! + 3^4*L(x)L(4x)L(16x)L(64x)/4! + ...
. 1/E(x) = 1 - L(x) + L(x)L(4x)/2! - L(x)L(4x)L(16x)/3! + L(x)L(4x)L(16x)L(64x)/4! -+ ...
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{a(n,q=4)=local(Lq=x+x^2);for(i=1,n,Lq=x*sum(m=0,n,(q-1)^m/m!*prod(k=0,m-1,subst(Lq,x,q^k*x+x*O(x^n)))));n!*polcoeff(Lq,n)}
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{a(n,q=4)=n!*polcoeff(sum(m=1,n,q^(m*(m-1)/2)*x^m/(m-1)!)/sum(m=0,n,q^(m*(m-1)/2)*x^m/m!+x*O(x^n)),n)} \\ Paul D. Hanna, Aug 31 2010
A177778
E.g.f.: A(x) = Sum_{n>=0} 2^n/n!*Product_{k=0..n-1} L(2^k*x), where L(x) is the e.g.f. of A177777.
Original entry on oeis.org
1, 2, 12, 160, 4272, 221648, 22347648, 4416360160, 1724182065408, 1336677590208512, 2064038664552586752, 6359502604300426739200, 39136760890428640414851072, 481344480930558145524346370048
Offset: 0
E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 160*x^3/3! + 4272*x^4/4! +...
Then e.g.f. A(x) is given by:
A(x) = 1 + 2*L(x) + 2^2*L(x)L(2x)/2! + 2^3*L(x)L(2x)L(4x)/3! + 2^4*L(x)L(2x)L(4x)L(8x)/4! +...
where L(x) is the e.g.f. of A177777:
. L(x) = x + 2*x^2/2! + 12*x^3/3! + 152*x^4/4! + 3640*x^5/5! +...
. L(x) = x*d/dx log( Sum_{n>=0} 2^(n(n-1)/2)*x^n/n! )
and satisfies:
. L(x)/x = 1 + L(x) + L(x)L(2x)/2! + L(x)L(2x)L(4x)/3! + L(x)L(2x)L(4x)L(8x)/4! +...
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{a(n,q=2)=local(Lq=x+x^2,A); for(i=1,n,Lq=x*sum(m=0,n,(q-1)^m/m!*prod(k=0,m-1,subst(Lq,x,q^k*x+x*O(x^n))))); A=sum(m=0,n,2^m/m!*prod(k=0,m-1,subst(Lq,x,q^k*x+x*O(x^n))));n!*polcoeff(A,n)}
Showing 1-3 of 3 results.
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