A296171
O.g.f. A(x) satisfies: [x^n] exp( n^2 * A(x) ) = [x^(n-1)] exp( n^2 * A(x) ) for n>=1.
Original entry on oeis.org
1, -1, -1, -9, -134, -2852, -79096, -2699480, -109201844, -5100872244, -269903909820, -15944040740604, -1039553309158964, -74123498185170292, -5736368141560365292, -478780244956262592748, -42865943103053965559668, -4097785410628237071311764, -416572537937169684523985420, -44873737158384968851319470220, -5106038963454360810619516396820, -611986780692307637617151164361140, -77066319756799442735378541663266476
Offset: 1
G.f. A(x) = x - x^2 - x^3 - 9*x^4 - 134*x^5 - 2852*x^6 - 79096*x^7 - 2699480*x^8 - 109201844*x^9 - 5100872244*x^10 - 269903909820*x^11 - 15944040740604*x^12 - 1039553309158964*x^13 - 74123498185170292*x^14 - 5736368141560365292*x^15 + ...
such that
G(x) = exp(A(x)) = 1 + x - x^2/2! - 11*x^3/3! - 239*x^4/4! - 17059*x^5/5! - 2145689*x^6/6! - 412595231*x^7/7! - 111962826751*x^8/8! - 40590007936199*x^9/9! - 18900753214178609*x^10/10! + ... + A296170(n)*x^n/n! + ...
satisfies [x^(n-1)] G(x)^(n^2) = [x^n] G(x)^(n^2) for n>=1.
RELATED SERIES.
Series_Reversion(A(x)) = x + x^2 + 3*x^3 + 19*x^4 + 226*x^5 + 4259*x^6 + 110514*x^7 + 3626207*x^8 + 143043592*x^9 + 6567931068*x^10 + 343278693103*x^11 + 20092744961109*x^12 + 1300754163383700*x^13 + ... + A295812(n)*x^n + ...
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{a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^2)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^2 ); polcoeff(log(Ser(A)),n)}
for(n=1,30,print1(a(n),", "))
A295813
G.f. A(x) satisfies: G(A(x)) = exp(x), where G(x) equals the e.g.f. of A296172.
Original entry on oeis.org
1, 3, 48, 3271, 575163, 185377116, 93039467356, 66505075585875, 63970743282062646, 79580632411431634441, 124299284968805234137968, 238188439678208173206500760, 549611050835556942751087049225, 1503700734638162443238902233252144, 4814751647416985610768723994195186728, 17841762828286483988438913318683740082187, 75777421917902616009655480827109144353730842
Offset: 1
G.f.: A(x) = x + 3*x^2 + 48*x^3 + 3271*x^4 + 575163*x^5 + 185377116*x^6 + 93039467356*x^7 + 66505075585875*x^8 + 63970743282062646*x^9 + 79580632411431634441*x^10 + 124299284968805234137968*x^11 + 238188439678208173206500760*x^12 +...
The series reversion equals the logarithm of the e.g.f. of A296172, which begins:
Series_Reversion(A(x)) = x - 3*x^2 - 30*x^3 - 2686*x^4 - 517311*x^5 - 173118807*x^6 - 88535206152*x^7 - 63977172334344*x^8 - 61971659588102940*x^9 - 77470793599569049440*x^10 - 121439997599825393413344*x^11 - 233353875172602479932391040*x^12 +...+ A296173(n)*x^n +...
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{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^3)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^3 ); polcoeff(serreverse(log(Ser(A))), n)}
for(n=1, 30, print1(a(n), ", "))
A295814
G.f. A(x) satisfies: G(A(x)) = exp(x), where G(x) equals the e.g.f. of A296174.
Original entry on oeis.org
1, 7, 591, 360071, 696409901, 2958728428011, 23164541753169117, 300801581861406441263, 6028093825088113213286946, 176753891171734450100762135773, 7275100380834838623971362431809230, 406542590169784279153263825042856310627, 30008177367626616771665421796780382440931316, 2859139755874441545650368872575815286528870509597
Offset: 1
G.f. A(x) = x + 7*x^2 + 591*x^3 + 360071*x^4 + 696409901*x^5 + 2958728428011*x^6 + 23164541753169117*x^7 + 300801581861406441263*x^8 +...
Series_Reversion(A(x)) = x - 7*x^2 - 493*x^3 - 341101*x^4 - 680813601*x^5 - 2923660883625*x^6 - 22996362478599551*x^7 - 299331006952284448127*x^8 - 6006951481145880962408552*x^9 +...+ A296175(n)*x^n +...
G(x) = exp(Series_Reversion(A(x))) = 1 + x - 13*x^2/2! - 2999*x^3/3! - 8197751*x^4/4! - 81738176899*x^5/5! - 2105524335759389*x^6/6! - 115916378979693710123*x^7/7! - 12069952631345502122877199*x^8/8! - 2179911119857340269414590758951*x^9/9! +...+ A296174(n)*x^n/n! +...
which satisfies [x^(n-1)] G(x)^(n^4) = [x^n] G(x)^(n^4) for n>=1.
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{a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^4)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^4 ); polcoeff(serreverse(log(Ser(A))),n)}
for(n=1,30,print1(a(n),", "))
Showing 1-3 of 3 results.
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