A301363
G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x)/(1 - 2*x*A(x)/(1 - 3*x*A(x)/(1 - 4*x*A(x)/(1 - ...))))), a continued fraction.
Original entry on oeis.org
1, 1, 4, 25, 202, 1966, 22306, 289969, 4272934, 70792318, 1308702592, 26791202362, 602762346088, 14795609964448, 393567982759966, 11276489767853569, 346158428070229414, 11331678979354212886, 393967314482937530248, 14495027742943618066030, 562600190990455844759356
Offset: 0
G.f. A(x) = 1 + x + 4*x^2 + 25*x^3 + 202*x^4 + 1966*x^5 + 22306*x^6 + 289969*x^7 + 4272934*x^8 + ...
The g.f. also satisfies:
A(x) = 1 + x*A(x) + 3*x^2*A(x) + 15*x^3*A(x)^3 + 105*x^4*A(x)^4 + 945*x^5*A(x)^5 + 10395*x^6*A(x)^6 + ... + (2*n)!/(n!*2^n) * x^n * A(x)^n + ...
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/* Continued Fraction */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1,n, A=CF; for(k=0, n, CF = 1/(1 - (n-k+1)*x*A*CF ) )); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", "))
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/* Differential Equation */
{a(n) = my(A=1); for(i=0,n, A = 1 + x*(x*A^3)'/(x*A +x^2*O(x^n))'); polcoeff(A,n)}
for(n=0, 30, print1(a(n),", "))
A302535
G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1).
Original entry on oeis.org
1, 1, 6, 61, 846, 14746, 310016, 7665141, 218827766, 7106293246, 259169817316, 10497928495506, 467768758203676, 22739720141372196, 1197560448125948596, 67910602688355999461, 4125144974025630599846, 267199960610924528490486, 18382741943990196237909476, 1338585578875261292134492646, 102848696213697953204782043556
Offset: 0
G.f.: A(x) = 1 + x + 6*x^2 + 61*x^3 + 846*x^4 + 14746*x^5 + 310016*x^6 + 7665141*x^7 + 218827766*x^8 + 7106293246*x^9 + 259169817316*x^10 + ...
such that
A(x) = 1 + x*A(x) + 5*x^2*A(x)^2 + 45*x^3*A(x)^3 + 585*x^4*A(x)^4 + 9945*x^5*A(x)^5 + 208845*x^6*A(x)^6 + ... + x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1) + ...
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/* Series Reversion of Quartic Factorials g.f.: */
{a(n) = polcoeff((1/x) * serreverse(x/sum(m=0, n, x^m*prod(k=1,m-1,4*k + 1))+x^2*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
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/* Differential Equation: */
{a(n) = my(A=1); for(i=0, n, A = 1 + x*A^2*(A + 5*x*A')/(x*A +x^2*O(x^n))'); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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/* Continued fraction: */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, A=CF; for(k=0, n, CF = 1/(1 - floor(4*floor(3*(n-k+1)/2)/3)*x*A*CF ) )); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", "))
A302565
G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (5*k + 1).
Original entry on oeis.org
1, 1, 7, 85, 1429, 30517, 792007, 24293389, 862902745, 34918162057, 1587910815271, 80217252865861, 4457823231346717, 270261899977497325, 17749585402744292215, 1255201826997862952845, 95083758340337074058545, 7680863233559647281837265, 659040900304099125516970375, 59855299015030039092312638965
Offset: 0
G.f.: A(x) = 1 + x + 7*x^2 + 85*x^3 + 1429*x^4 + 30517*x^5 + 792007*x^6 + 24293389*x^7 + 862902745*x^8 + 34918162057*x^9 + ...
such that
A(x) = 1 + x*A(x) + 6*x^2*A(x)^2 + 66*x^3*A(x)^3 + 1056*x^4*A(x)^4 + 22176*x^5*A(x)^5 + ... + x^n*A(x)^n * Product_{k=0..n-1} (5*k + 1) + ...
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/* Series Reversion of Quintuple Factorials g.f.: */
{a(n) = polcoeff((1/x) * serreverse(x/sum(m=0, n, x^m * prod(k=0, m-1, 5*k + 1)) +x^2*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
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/* Differential Equation: */
{a(n) = my(A=1); for(i=0, n, A = 1 + x*A^2*(A + 6*x*A')/(x*A +x^2*O(x^n))'); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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/* Continued fraction: */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, A=CF; for(k=0, n, CF = 1/(1 - floor(5*floor(3*(n-k+1)/2)/3)*x*A*CF ) )); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.