A301581
G.f.: Sum_{n>=0} 2^n * ((1+x)^n - 1)^n.
Original entry on oeis.org
1, 2, 16, 232, 4748, 125440, 4058312, 155336672, 6864980968, 343995674280, 19270975801600, 1193481831243584, 80966964261458368, 5971270693661978816, 475655179279901897536, 40699219246551726635840, 3722813577249648564213392, 362519587815189751405383520, 37442485808471509306691295808, 4088344078912544484116541775616, 470550859964811044524886252649760
Offset: 0
G.f.: A(x) = 1 + 2*x + 16*x^2 + 232*x^3 + 4748*x^4 + 125440*x^5 + 4058312*x^6 + 155336672*x^7 + 6864980968*x^8 + 343995674280*x^9 + ...
such that
A(x) = 1 + 2*((1+x)-1) + 4*((1+x)^2-1)^2 + 8*((1+x)^3-1)^3 + 16*((1+x)^4-1)^4 + 32*((1+x)^5-1)^5 + 64*((1+x)^6-1)^6 + 128*((1+x)^7-1)^7 + ...
Also,
A(x) = 1/3 + 2*(1+x)/(1 + 2*(1+x))^2 + 4*(1+x)^4/(1 + 2*(1+x)^2)^3 + 8*(1+x)^9/(1 + 2*(1+x)^3)^4 + 16*(1+x)^16/(1 + 2*(1+x)^4)^5 + 32*(1+x)^25/(1 + 2*(1+x)^5)^6 + ...
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nmax = 20; CoefficientList[Series[1 + Sum[2^j*((1 + x)^j - 1)^j, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2020 *)
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{a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, 2^m * ((1+x +o)^m - 1)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
A301582
G.f.: Sum_{n>=0} 3^n * ((1+x)^n - 1)^n.
Original entry on oeis.org
1, 3, 36, 765, 22932, 886707, 41971041, 2349915543, 151893243711, 11131097539221, 911906584505874, 82586031357156975, 8192750710914222984, 883506535094875209327, 102907862475072248379060, 12875067336646598300376165, 1722014444866824121524712497, 245185575019136812676809863351, 37027348593726417935247243009495, 5911490521308027393188499233189367, 994821814352463817234026392636083551
Offset: 0
G.f.: A(x) = 1 + 3*x + 36*x^2 + 765*x^3 + 22932*x^4 + 886707*x^5 + 41971041*x^6 + 2349915543*x^7 + 151893243711*x^8 + ...
such that
A(x) = 1 + 3*((1+x)-1) + 9*((1+x)^2-1)^2 + 27*((1+x)^3-1)^3 + 81*((1+x)^4-1)^4 + 243*((1+x)^5-1)^5 + 729*((1+x)^6-1)^6 + 2187*((1+x)^7-1)^7 + ...
Also,
A(x) = 1/4 + 3*(1+x)/(1 + 3*(1+x))^2 + 9*(1+x)^4/(1 + 3*(1+x)^2)^3 + 27*(1+x)^9/(1 + 3*(1+x)^3)^4 + 81*(1+x)^16/(1 + 3*(1+x)^4)^5 + 243*(1+x)^25/(1 + 3*(1+x)^5)^6 + ...
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nmax = 20; CoefficientList[Series[1 + Sum[3^j*((1 + x)^j - 1)^j, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2020 *)
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{a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, 3^m * ((1+x +o)^m - 1)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
A301583
G.f.: Sum_{n>=0} 4^n * ((1+x)^n - 1)^n.
Original entry on oeis.org
1, 4, 64, 1792, 70736, 3600128, 224255040, 16521605376, 1405131880000, 135480346104896, 14602769310474240, 1739917222954854400, 227081534040721917952, 32217108743091290851328, 4936803887495636263284736, 812576030237749532251019264, 142976863303365903802301729024, 26781577193841845859144244087808, 5320767287406003709062843236972544, 1117525692987087894816123931091214336
Offset: 0
G.f.: A(x) = 1 + 4*x + 64*x^2 + 1792*x^3 + 70736*x^4 + 3600128*x^5 + 224255040*x^6 + 16521605376*x^7 + 1405131880000*x^8 + ...
such that
A(x) = 1 + 4*((1+x)-1) + 16*((1+x)^2-1)^2 + 64*((1+x)^3-1)^3 + 256*((1+x)^4-1)^4 + 1024*((1+x)^5-1)^5 + 4096*((1+x)^6-1)^6 + ...
Also,
A(x) = 1/5 + 4*(1+x)/(1 + 4*(1+x))^2 + 16*(1+x)^4/(1 + 4*(1+x)^2)^3 + 64*(1+x)^9/(1 + 4*(1+x)^3)^4 + 256*(1+x)^16/(1 + 4*(1+x)^4)^5 + 1024*(1+x)^25/(1 + 4*(1+x)^5)^6 + ...
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nmax = 20; CoefficientList[Series[1 + Sum[4^j*((1 + x)^j - 1)^j, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2020 *)
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{a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, 4^m * ((1+x +o)^m - 1)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
A317798
G.f.: Sum_{n>=0} (3*(1+x)^n - 1)^n / 3^(n+1).
Original entry on oeis.org
1, 15, 786, 69261, 8554530, 1359020643, 263929299177, 60582032629791, 16046282916588207, 4817035600778756553, 1616224504900354928832, 599373591433178971787007, 243449152911402772344286998, 107482020677618238226506065235, 51249638236281451846248205583562, 26247197050200652206165329786055981, 14369481728948627418149559363836673273
Offset: 0
G.f.: A(x) = 1 + 15*x + 786*x^2 + 69261*x^3 + 8554530*x^4 + 1359020643*x^5 + 263929299177*x^6 + 60582032629791*x^7 + 16046282916588207*x^8 + ...
such that
A(x) = 1/3 + (3*(1+x) - 1)/3^2 + (3*(1+x)^2 - 1)^3/3^3 + (3*(1+x)^3 - 1)^3/3^4 + (3*(1+x)^4 - 1)^4/3^5 + (3*(1+x)^5 - 1)^5/3^6 + ...
Also,
A(x) = 1/4 + 3*(1+x)/(3 + (1+x))^2 + 3^2*(1+x)^4/(3 + (1+x)^2)^3 + 3^3*(1+x)^9/(3 + (1+x)^3)^4 + 3^4*(1+x)^16/(3 + (1+x)^4)^5 + 3^5*(1+x)^25/(3 + (1+x)^5)^6 + 3^6*(1+x)^36/(3 + (1+x)^6)^7 + ...
A317799
G.f.: Sum_{n>=0} (4*(1+x)^n - 1)^n / 4^(n+1).
Original entry on oeis.org
1, 28, 2644, 418108, 92624756, 26388012380, 9189259388052, 3782063138596476, 1796136011427955636, 966755321167565129372, 581573928178258915024596, 386690499153558305585430460, 281600848152507182372274325492, 222904650325844057584524049181660, 190559248618061561787517993382005012
Offset: 0
G.f.: A(x) = 1 + 28*x + 2644*x^2 + 418108*x^3 + 92624756*x^4 + 26388012380*x^5 + 9189259388052*x^6 + 3782063138596476*x^7 + 1796136011427955636*x^8 + ...
such that
A(x) = 1/4 + (4*(1+x) - 1)/4^2 + (4*(1+x)^2 - 1)^3/4^3 + (4*(1+x)^3 - 1)^4/4^4 + (4*(1+x)^4 - 1)^4/4^5 + (4*(1+x)^5 - 1)^5/4^6 + ...
Also,
A(x) = 1/5 + 4*(1+x)/(4 + (1+x))^2 + 4^2*(1+x)^4/(4 + (1+x)^2)^4 + 4^3*(1+x)^9/(4 + (1+x)^3)^4 + 4^4*(1+x)^16/(4 + (1+x)^4)^5 + 4^5*(1+x)^25/(4 + (1+x)^5)^6 + 4^6*(1+x)^36/(4 + (1+x)^6)^7 + ...
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