A319947
G.f.: Sum_{n>=0} ( 1/(1-x)^n - (1-x)^n )^n.
Original entry on oeis.org
1, 2, 17, 233, 4457, 109599, 3294200, 117023348, 4796944724, 222859320409, 11572143728964, 664158801170094, 41748985785588788, 2852580634624308469, 210503045435437702457, 16684642612290860954017, 1413651317086090261964496, 127503642994522759923638691, 12197174216389125259958117521, 1233478106868364650369933771887
Offset: 0
G.f.: A(x) = 1 + 2*x + 17*x^2 + 233*x^3 + 4457*x^4 + 109599*x^5 + 3294200*x^6 + 117023348*x^7 + 4796944724*x^8 + 222859320409*x^9 + ...
such that
A(x) = 1 + (1/(1-x) - (1-x)) + (1/(1-x)^2 - (1-x)^2)^2 + (1/(1-x)^3 - (1-x)^3)^3 + (1/(1-x)^4 - (1-x)^4)^4 + (1/(1-x)^5 - (1-x)^5)^5 + ...
Equivalently,
A(x) = 1 +
(1/(1-x) - (1-x)) +
(1/(1-x)^4 - 2 + (1-x)^4) +
(1/(1-x)^9 - 3/(1-x)^3 + 3*(1-x)^3 - (1-x)^9) +
(1/(1-x)^16 - 4/(1-x)^8 + 6 - 4*(1-x)^8 + (1-x)^16) +
(1/(1-x)^25 - 5/(1-x)^15 + 10/(1-x)^5 - 10*(1-x)^5 + 5*(1-x)^15 - (1-x)^25) +
(1/(1-x)^36 - 6/(1-x)^24 + 15/(1-x)^12 - 20 + 15*(1-x)^12 - 6*(1-x)^24 + (1-x)^36) +
...
-
{a(n) = my(A=1, X=x + x*O(x^n)); A = sum(m=0,n, (1/(1-X)^m - (1-x)^m)^m );polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
A369088
Expansion of Sum_{n>=0} ( (1+x)^n/(1-x)^n - (1-x)^n/(1+x)^n )^n / 4^n.
Original entry on oeis.org
1, 1, 4, 28, 280, 3639, 57996, 1093795, 23821104, 588282772, 16243898516, 495894495629, 16584179388232, 602955889304341, 23678788166350620, 998882687260157956, 45047554811998482016, 2162775743390757357579, 110136661581764181626660, 5929361362606879245799055, 336484778280758295928357240
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 280*x^4 + 3639*x^5 + 57996*x^6 + 1093795*x^7 + 23821104*x^8 + 588282772*x^9 + 16243898516*x^10 + ...
By definition, A(x) equals the sum
A(x) = 1 + B_1(x) + B_2(x)^2 + B_3(x)^3 + B_4(x)^4 + ... + B_n(x)^n + ...
where
B_n(x) = ( (1+x)^n/(1-x)^n - (1-x)^n/(1+x)^n )/4,
explicitly,
B_1(x) = x + x^3 + x^5 + x^7 + x^9 + ...
B_2(x) = 2*x + 6*x^3 + 10*x^5 + 14*x^7 + 18*x^9 + ...
B_3(x) = 3*x + 19*x^3 + 51*x^5 + 99*x^7 + 163*x^9 + ...
B_4(x) = 4*x + 44*x^3 + 180*x^5 + 476*x^7 + 996*x^9 + ...
B_5(x) = 5*x + 85*x^3 + 501*x^5 + 1765*x^7 + 4645*x^9 + ...
B_6(x) = 6*x + 146*x^3 + 1182*x^5 + 5418*x^7 + 17718*x^9 + ...
B_7(x) = 7*x + 231*x^3 + 2471*x^5 + 14407*x^7 + 57799*x^9 + ...
...
and
B_2(x)^2 = 4*x^2 + 24*x^4 + 76*x^6 + 176*x^8 + 340*x^10 + ...
B_3(x)^3 = 27*x^3 + 513*x^5 + 4626*x^7 + 26974*x^9 + 116901*x^11 + ...
B_4(x)^4 = 256*x^4 + 11264*x^6 + 231936*x^8 + 3005440*x^10 + ...
B_5(x)^5 = 3125*x^5 + 265625*x^7 + 10596875*x^9 + 265509375*x^11 + ...
B_6(x)^6 = 46656*x^6 + 6811776*x^8 + 469530432*x^10 + ...
B_7(x)^7 = 823543*x^7 + 190238433*x^9 + 20868579620*x^11 + ...
...
-
{a(n) = my(A=1, X=x + x*O(x^n)); A = sum(m=0, n, ((1+X)^m/(1-X)^m - (1-X)^m/(1+X)^m)^m/4^m ); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
A320419
E.g.f.: Sum_{n>=0} 2^n * sinh(n*x)^n.
Original entry on oeis.org
1, 2, 32, 1298, 98816, 12116642, 2181373952, 541793612978, 177515752718336, 74174630255081282, 38495436789222735872, 24292625097918019749458, 18317925825330618728185856, 16266073932645598088605425122, 16800468023465020621665905672192, 19969924961381649826994229325322738
Offset: 0
E.g.f.: A(x) = 1 + 2*x + 32*x^2/2! + 1298*x^3/3! + 98816*x^4/4! + 12116642*x^5/5! + 2181373952*x^6/6! + 541793612978*x^7/7! + ...
such that
A(x) = 1 + 2*sinh(x) + 4*sinh(2*x)^2 + 8*sinh(3*x)^3 + 16*sinh(4*x)^4 +...
or, equivalently,
A(x) = 1 + exp(x)*(1 - exp(-2*x)) + exp(4*x)*(1 - exp(-4*x))^2 + exp(9*x)*(1 - exp(-6*x))^3 + exp(16*x)*(1 - exp(-8*x))^4 + exp(25*x)*(1 - exp(-10*x))^5 + ...
RELATED SERIES.
A(log(1+x)) = 1 + 2*x + 15*x^2 + 201*x^3 + 3807*x^4 + 93103*x^5 + 2788528*x^6 + 98816388*x^7 + 4043274742*x^8 + ... + A319466(n)*x^n + ...
A(-log(1-x)) = 1 + 2*x + 17*x^2 + 233*x^3 + 4457*x^4 + 109599*x^5 + 3294200*x^6 + 117023348*x^7 + 4796944724*x^8 + ... + A319947(n)*x^n + ...
-
{a(n) = n! * polcoeff(sum(k=0, n, 2^k * sinh(k*x + x*O(x^n))^k ), n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.
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