A363305
Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^5 + A(x)^9).
Original entry on oeis.org
1, 2, 28, 576, 13968, 371280, 10465152, 307252032, 9295409664, 287758274304, 9071667965184, 290237226038272, 9399819302979584, 307570021821937664, 10152439243763290112, 337658352835320934400, 11304320019217804476416, 380650592731460987617280
Offset: 0
G.f.: A(x) = 1 + 2*x + 28*x^2 + 576*x^3 + 13968*x^4 + 371280*x^5 + 10465152*x^6 + 307252032*x^7 + 9295409664*x^8 + ...
where A(x) = 1 + x*(A(x)^5 + A(x)^9).
RELATED SERIES.
A(x)^5 = 1 + 10*x + 180*x^2 + 4080*x^3 + 104160*x^4 + 2858352*x^5 + 82336320*x^6 + 2455727040*x^7 + ...
A(x)^9 = 1 + 18*x + 396*x^2 + 9888*x^3 + 267120*x^4 + 7606800*x^5 + 224915712*x^6 + 6839682624*x^7 + ...
-
{a(n) = sum(k=0, n, binomial(n, k)*binomial(5*n+4*k+1, n)/(5*n+4*k+1) )}
for(n=0, 20, print1(a(n), ", "))
A363380
G.f. satisfies A(x) = 1 + x * A(x)^4 * (1 + A(x)^2).
Original entry on oeis.org
1, 2, 20, 284, 4712, 85392, 1638112, 32699472, 672188768, 14133399744, 302535052160, 6570819330688, 144442463464704, 3207564324825600, 71848240540852224, 1621452789508328704, 36831997860270007808, 841470878382566444032
Offset: 0
-
a(n) = sum(k=0, n, binomial(n, k)*binomial(4*n+2*k+1, n)/(4*n+2*k+1));
A364400
G.f. satisfies A(x) = 1 + x/A(x)^3*(1 + 1/A(x)^3).
Original entry on oeis.org
1, 2, -18, 270, -4902, 98538, -2110794, 47227846, -1090742094, 25806364434, -622267199554, 15236456140542, -377814588773622, 9468373002766074, -239434464005544570, 6101951612867546166, -156561081975745809566, 4040863076496835880226
Offset: 0
-
a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(4*n+3*k-2, n-1))/n);
A363111
Expansion of g.f. A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.
Original entry on oeis.org
1, 1, 11, 127, 1547, 19652, 258069, 3481034, 47999915, 674086924, 9612919156, 138878011335, 2028718584989, 29918897595468, 444889269572286, 6663228661354420, 100430376524360459, 1522215623202615036, 23187346871707554564, 354783440893854307244
Offset: 0
G.f.: A(x) = 1 + x + 11*x^2 + 127*x^3 + 1547*x^4 + 19652*x^5 + 258069*x^6 + 3481034*x^7 + 47999915*x^8 + 674086924*x^9 + ...
such that A(x) = F(x*F(x)^7) where F(x) = 1 + x*F(x)^4 begins
F(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 + 53820*x^7 + ... + A002293(n)*x^n + ...
RELATED SERIES.
Let B(x) = A(x*B(x)^3) = ( Series_Reversion( x/A(x)^3 )/x )^(1/3) which begins
B(x) = 1 + x + 14*x^2 + 238*x^3 + 4578*x^4 + 95130*x^5 + 2082150*x^6 + 47295990*x^7 + 1104598378*x^8 + ...
then
( (B(x) - 1)/x )^(1/7) = 1 + 2*x + 22*x^2 + 350*x^3 + 6538*x^4 + 133658*x^5 + 2895214*x^6 + 65294502*x^7 + ... + A363304(n)*x^n + ...
is an integer series.
-
{a(n) = if(n==0, 1, sum(k=1, n, 7*k* binomial(4*k+1, k) * binomial(4*n+3*k, n-k) / ((4*k+1)*(4*n+3*k)) ) )}
for(n=0, 30, print1(a(n), ", "))
-
/* G.f. A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4 */
{a(n) = my(F = 1); for(i=1,n, F = 1 + x*F^4 + x*O(x^n));
polcoeff( subst(F, x, x*F^7), n)}
for(n=0, 30, print1(a(n), ", "))
A364195
Expansion of g.f. A(x) satisfying A(x) = 1 + x * A(x)^5 * (1 + A(x)^2).
Original entry on oeis.org
1, 2, 24, 412, 8280, 181904, 4232048, 102479184, 2555884896, 65207430848, 1693785940992, 44643489969792, 1190986788639232, 32097745138518528, 872595854798515456, 23900545715576753408, 658934625866433496576, 18271554709525993556992, 509241947434834351042560
Offset: 0
-
a(n) = sum(k=0, n, binomial(n, k)*binomial(5*n+2*k+1, n)/(5*n+2*k+1));
Showing 1-5 of 5 results.
Comments