A248653
E.g.f.: Sum_{n>=0} x^n * (2 + exp(n*x))^n.
Original entry on oeis.org
1, 3, 20, 237, 4276, 107225, 3518526, 145005721, 7285611096, 436297841649, 30590014543930, 2474931380486081, 228308751882636756, 23772216923031342649, 2769853988736186166374, 358463639909150646730665, 51192480930691715108562736, 8021370202848006225125239649
Offset: 0
G.f.: A(x) = 1 + 3*x + 20*x^2/2! + 237*x^3/3! + 4276*x^4/4! + 107225*x^5/5! +...
where the g.f. satisfies following series identity:
A(x) = 1 + x*(2+exp(x)) + x^2*(2+exp(2*x))^2 + x^3*(2+exp(3*x))^3 + x^4*(2+exp(4*x))^4 + x^5*(2+exp(5*x))^5 + x^6*(2+exp(6*x))^6 +...
A(x) = 1/(1-2*x) + x*exp(x)/(1-2*x*exp(x))^2 + x^2*exp(4*x)/(1-2*x*exp(2*x))^3 + x^3*exp(9*x)/(1-2*x*exp(3*x))^4 + x^4*exp(16*x)/(1-2*x*exp(4*x))^5 + x^5*exp(25*x)/(1-2*x*exp(5*x))^6 + x^6*exp(36*x)/(1-2*x*exp(6*x))^7 +...
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{a(n,t=2)=local(A=1); A=sum(k=0, n, x^k * (t + exp(k*x +x*O(x^n)))^k); n!*polcoeff(A, n)}
for(n=0,25,print1(a(n,2),", "))
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{a(n,t=2)=local(A=1); A=sum(k=0, n, x^k * exp(k^2*x +x*O(x^n))/(1 - t*x*exp(k*x +x*O(x^n)))^(k+1) ); n!*polcoeff(A, n)}
for(n=0,25,print1(a(n,2),", "))
A248654
E.g.f.: Sum_{n>=0} x^n * (3 + exp(n*x))^n.
Original entry on oeis.org
1, 4, 34, 483, 10084, 286885, 10556406, 483876799, 26866889512, 1768601369961, 135698985275050, 11968589697570451, 1199598182911257372, 135313142875442335453, 17035239326998414091038, 2376497634554143028502855, 365070055205852728328220496, 61412309543674687202717299921
Offset: 0
G.f.: A(x) = 1 + 4*x + 34*x^2/2! + 483*x^3/3! + 10084*x^4/4! + 286885*x^5/5! +...
where the g.f. satisfies following series identity:
A(x) = 1 + x*(3+exp(x)) + x^2*(3+exp(2*x))^2 + x^3*(3+exp(3*x))^3 + x^4*(3+exp(4*x))^4 + x^5*(3+exp(5*x))^5 + x^6*(3+exp(6*x))^6 +...
A(x) = 1/(1-3*x) + x*exp(x)/(1-3*x*exp(x))^2 + x^2*exp(4*x)/(1-3*x*exp(2*x))^3 + x^3*exp(9*x)/(1-3*x*exp(3*x))^4 + x^4*exp(16*x)/(1-3*x*exp(4*x))^5 + x^5*exp(25*x)/(1-3*x*exp(5*x))^6 + x^6*exp(36*x)/(1-3*x*exp(6*x))^7 +...
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{a(n,t=3)=local(A=1); A=sum(k=0, n, x^k * exp(k^2*x +x*O(x^n))/(1 - t*x*exp(k*x +x*O(x^n)))^(k+1) ); n!*polcoeff(A, n)}
for(n=0,25,print1(a(n,3),", "))
A248655
E.g.f.: Sum_{n>=0} x^n * (2*exp(n*x) - 1)^n.
Original entry on oeis.org
1, 1, 6, 60, 1040, 24650, 796332, 32556650, 1650306352, 100688420034, 7262453789300, 609263939139482, 58684327404118440, 6419194423681262210, 790005709370352467020, 108514436287885023435930, 16520198768574933272377952, 2770418328864442110268310402, 508995643667831746279224633828
Offset: 0
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 60*x^3/3! + 1040*x^4/4! + 24650*x^5/5! +...
where the e.g.f. satisfies following series identity:
A(x) = 1 + x*(2*exp(x)-1) + x^2*(2*exp(2*x)-1)^2 + x^3*(2*exp(3*x)-1)^3 + x^4*(2*exp(4*x)-1)^4 + x^5*(2*exp(5*x)-1)^5 + x^6*(2*exp(6*x)-1)^6 +...
A(x) = 1/(1+x) + (2*x)*exp(x)/(1+x*exp(x))^2 + (2*x)^2*exp(4*x)/(1+x*exp(2*x))^3 + (2*x)^3*exp(9*x)/(1+x*exp(3*x))^4 + (2*x)^4*exp(16*x)/(1+x*exp(4*x))^5 + (2*x)^5*exp(25*x)/(1+x*exp(5*x))^6 + (2*x)^6*exp(36*x)/(1+x*exp(6*x))^7 +...
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{a(n)=local(A=1); A=sum(k=0, n, x^k * (2*exp(k*x +x*O(x^n)) - 1)^k); n!*polcoeff(A, n)}
for(n=0,25,print1(a(n),", "))
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{a(n)=local(A=1); A=sum(k=0, n, (2*x)^k * exp(k^2*x +x*O(x^n)) / (1 + x*exp(k*x +x*O(x^n)))^(k+1) ); n!*polcoeff(A, n)}
for(n=0,25,print1(a(n),", "))
A249409
E.g.f.: Sum_{n>=0} x^n * (exp(n*x) - 1)^n.
Original entry on oeis.org
1, 0, 2, 3, 100, 965, 26166, 652687, 22791336, 942744105, 46394042410, 2686920755651, 179344249073628, 13675378594960381, 1179389058183817950, 114010570608973463895, 12267159489256485080656, 1459502985288809274179153, 190910980556047614953376594, 27316077853047666256173322603
Offset: 0
E.g.f.: A(x) = 1 + 2*x^2/2! + 3*x^3/3! + 100*x^4/4! + 965*x^5/5! +...
where the e.g.f. satisfies following series identity:
A(x) = 1 + x*(exp(x)-1) + x^2*(exp(2*x)-1)^2 + x^3*(exp(3*x)-1)^3 + x^4*(exp(4*x)-1)^4 + x^5*(exp(5*x)-1)^5 + x^6*(exp(6*x)-1)^6 +...
A(x) = 1/(1+x) + x*exp(x)/(1+x*exp(x))^2 + x^2*exp(4*x)/(1+x*exp(2*x))^3 + x^3*exp(9*x)/(1+x*exp(3*x))^4 + x^4*exp(16*x)/(1+x*exp(4*x))^5 + x^5*exp(25*x)/(1+x*exp(5*x))^6 + x^6*exp(36*x)/(1+x*exp(6*x))^7 +...
A(x) = 1/(x+1) + x*exp(-x)/(x+exp(-x))^2 + x^2*exp(-2*x)/(x+exp(-2*x))^3 + x^3*exp(-3*x)/(x+exp(-3*x))^4 + x^4*exp(-4*x)/(x+exp(-4*x))^5 + x^5*exp(-5*x)/(x+exp(-5*x))^6 + x^6*exp(-6*x)/(x+exp(-6*x))^7 +...
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{a(n)=local(A=1); A=sum(k=0, n, x^k * (exp(k*x +x*O(x^n)) - 1)^k); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
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{a(n)=local(A=1); A=sum(k=0, n, x^k * exp(k^2*x +x*O(x^n)) / (1 + x*exp(k*x +x*O(x^n)))^(k+1) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
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{a(n)=local(A=1); A=sum(k=0, n, x^k * exp(-k*x +x*O(x^n)) / (x + exp(-k*x +x*O(x^n)))^(k+1) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
A248471
E.g.f.: Sum_{n>=0} x^n * (3*exp(n*x) - 2)^n.
Original entry on oeis.org
1, 1, 8, 87, 1836, 49515, 1888218, 89221755, 5280079752, 373918542867, 31364236227630, 3055178634627603, 341669063554284348, 43366411236710160651, 6190420280148645465330, 985839231828100191119595, 173934377807560257682643088, 33790653299040390831598566051, 7189209864606264796455405368022
Offset: 0
E.g.f.: A(x) = 1 + x + 8*x^2/2! + 87*x^3/3! + 1836*x^4/4! + 49515*x^5/5! +...
where the e.g.f. satisfies following series identity:
A(x) = 1 + x*(3*exp(x)-2) + x^2*(3*exp(2*x)-2)^2 + x^3*(3*exp(3*x)-2)^3 + x^4*(3*exp(4*x)-2)^4 + x^5*(3*exp(5*x)-2)^5 + x^6*(3*exp(6*x)-2)^6 +...
A(x) = 1/(1+2*x) + (3*x)*exp(x)/(1+2*x*exp(x))^2 + (3*x)^2*exp(4*x)/(1+2*x*exp(2*x))^3 + (3*x)^3*exp(9*x)/(1+2*x*exp(3*x))^4 + (3*x)^4*exp(16*x)/(1+2*x*exp(4*x))^5 + (3*x)^5*exp(25*x)/(1+2*x*exp(5*x))^6 + (3*x)^6*exp(36*x)/(1+2*x*exp(6*x))^7 +...
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{a(n)=local(A=1); A=sum(k=0, n, x^k * (3*exp(k*x +x*O(x^n)) - 2)^k); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
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{a(n)=local(A=1); A=sum(k=0, n, (3*x)^k * exp(k^2*x +x*O(x^n)) / (1 + 2*x*exp(k*x +x*O(x^n)))^(k+1) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
Showing 1-5 of 5 results.