A225293
G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^n / Product_{k=1..n} (1 - k*x).
Original entry on oeis.org
1, 1, 3, 11, 46, 210, 1022, 5232, 27954, 155142, 892007, 5306785, 32662475, 208108337, 1374219242, 9418564346, 67102315232, 497617712664, 3844733673180, 30960923835040, 259797722635505, 2269726236363395, 20618932709111866, 194452174592422916, 1900387863379327247
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 46*x^4 + 210*x^5 + 1022*x^6 +...
where
A(x) = 1 + x*A(x)/(1-x) + x^2*A(x)^2/((1-x)*(1-2*x)) + x^3*A(x)^3/((1-x)*(1-2*x)*(1-3*x)) + x^4*A(x)^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) +...
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{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*A^m/prod(k=1, m, 1-k*x +x*O(x^n)) )); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
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{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=local(A=1+x);for(i=0,n,A=sum(m=0,n,x^m*sum(k=0,m,Stirling2(m,k)*(A+x*O(x^n))^k)));polcoeff(A,n)}
for(n=0, 20, print1(a(n), ", "))
A225294
G.f. satisfies: A(x) = Sum_{n>=0} x^n / Product_{k=1..n} (1 - k*x*A(x)).
Original entry on oeis.org
1, 1, 2, 6, 22, 92, 424, 2112, 11236, 63360, 376800, 2355016, 15430784, 105797968, 757866592, 5664174736, 44109816528, 357447744576, 3010091812000, 26304829992224, 238217024498432, 2232483865359488, 21621812897089536, 216130222764401024, 2226983944005048960
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 92*x^5 + 424*x^6 + 2112*x^7 +...
where
A(x) = 1 + x/(1-x*A(x)) + x^2/((1-x*A(x))*(1-2*x*A(x))) + x^3/((1-x*A(x))*(1-2*x*A(x))*(1-3*x*A(x))) + x^4/((1-x*A(x))*(1-2*x*A(x))*(1-3*x*A(x))*(1-4*x*A(x))) +...
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{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m/prod(k=1, m, 1-k*x*A +x*O(x^n)) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=local(A=1+x);for(i=0,n,A=sum(m=0,n,x^m*sum(k=0,m,Stirling2(m,k)*(A+x*O(x^n))^(m-k))));polcoeff(A,n)}
for(n=0, 20, print1(a(n), ", "))
A307402
G.f. A(x) satisfies: A(x) = Sum_{j>=0} j!*x^j*A(x)^j / Product_{k=1..j} (1 - k*x*A(x)).
Original entry on oeis.org
1, 1, 4, 23, 164, 1362, 12792, 133891, 1550148, 19772030, 277054232, 4252637446, 71248226536, 1297226168708, 25542157054944, 541131735552507, 12275049552454916, 296787898215881990, 7617196890240489912, 206772478080888288082, 5917589117194665548600, 178040033221054576103036
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 164*x^4 + 1362*x^5 + 12792*x^6 + 133891*x^7 + 1550148*x^8 + 19772030*x^9 + 277054232*x^10 + ...
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terms = 22; A[] = 1; Do[A[x] = Sum[j! x^j A[x]^j/Product[(1 - k x A[x]), {k, 1, j}], {j, 0, i}] + O[x]^i, {i, 1, terms}]; CoefficientList[A[x], x]
terms = 22; A[] = 1; Do[A[x] = Sum[(1/2) HurwitzLerchPhi[1/2, -k, 0] x^k A[x]^k, {k, 0, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 22; CoefficientList[1/x InverseSeries[Series[x/Sum[(1/2) HurwitzLerchPhi[1/2, -k, 0] x^k, {k, 0, terms}], {x, 0, terms}], x], x]
A301770
G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x) - x^2*A(x)^2/(1 - x*A(x) - 2*x^2*A(x)^2/(1 - x*A(x) - 3*x^2*A(x)^2/(1 - ...)))), a continued fraction.
Original entry on oeis.org
1, 1, 3, 11, 47, 217, 1061, 5399, 28337, 152381, 835823, 4660779, 26357111, 150872165, 872878665, 5098306063, 30034591105, 178326873753, 1066472979083, 6421120346267, 38907397325295, 237182461204097, 1454326514077709, 8968048205494983, 55608797571427793, 346716786105033077
Offset: 0
G.f. A(x) = 1 + x + 3*x^2 + 11*x^3 + 47*x^4 + 217*x^5 + 1061*x^6 + 5399*x^7 + 28337*x^8 + ...
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Table[SeriesCoefficient[(1 + Sum[(I/Sqrt[2])^k * HermiteH[k, -I/Sqrt[2]] * x^k, {k, 1, n}])^(n+1)/(n+1), {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Nov 05 2021 *)
Showing 1-4 of 4 results.
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