Original entry on oeis.org
2, 23, 292, 4068, 62861, 1075562, 20275944, 418724047, 9418874022, 229535650138, 6029910590473, 169978358728536, 5120224516689050, 164192864744507615, 5585978669387706724, 200988595873323113508, 7626780547253339197109, 304431018444668653351250, 12752396616363975496913952, 559388848572350998030227895
Offset: 2
-
A291843_ser(N, t='t) = {
my(x='x+O('x^N), y=1, y1=0, n=1,
dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
while (n++,
y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
(t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
if (y1 == y, break); y = y1; ); y;
};
A291844_ser(N, t='t) = {
my(z = A291843_ser(N+1, t));
((1+x)*z - 1)*(1 + t*x)/((1-t + t*(1+x)*z)*x*z^2);
};
A291844_kol(k, N=20) = {
my(s = A291844_ser(N+1+3*k\2, t='t + O('t^(k+1))));
Ser(polcoeff(s, k,'t), 'x, N);
};
Vec(A291844_kol(1))
Original entry on oeis.org
1, 1, 4, 29, 274, 3145, 42294, 651227, 11295242, 217954807, 4632600152, 107572674851, 2710093290348, 73635362430373, 2146667383237600, 66842659455306545, 2214207331808233390, 77752904712525291757, 2885163506590219810722, 112808263228714108970879
Offset: 0
-
A291843_ser(N, t='t) = {
my(x='x+O('x^N), y=1, y1=0, n=1,
dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
while (n++,
y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
(t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
if (y1 == y, break); y = y1; ); y;
};
A291844_ser(N, t='t) = {
my(z = A291843_ser(N+1, t));
((1+x)*z - 1)*(1 + t*x)/((1-t + t*(1+x)*z)*x*z^2);
};
A291844_kol(k, N=20) = {
my(s = A291844_ser(N+1+3*k\2, t='t + O('t^(k+1))));
Ser(polcoeff(s, k,'t), 'x, N);
};
Vec(A291844_kol(0))
A294159
Alternating row sums of triangle A291844.
Original entry on oeis.org
1, 1, 2, 6, 18, 55, 171, 538, 1708, 5461, 17560, 56728, 183973, 598597, 1953145, 6388376, 20939664, 68764283, 226192964, 745146462, 2458020664, 8118111977, 26841209903, 88835163150, 294284206183, 975699571009, 3237456793478, 10749922312752, 35718863630895, 118757413662397
Offset: 0
-
A291843_ser(N, t='t) = {
my(x='x+O('x^N), y=1, y1=0, n=1,
dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
while (n++,
y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
(t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
if (y1 == y, break); y = y1; ); y;
};
A291844_ser(N, t='t) = {
my(z = A291843_ser(N+1, t));
((1+x)*z - 1)*(1 + t*x)/((1-t + t*(1+x)*z)*x*z^2);
};
Vec(A291844_ser(30,-1))
\\ test: y=A291844_ser(200,-1); 0==(x^3 + x^2 + 3*x - 1)*(y^2 - y) + x
Original entry on oeis.org
1, 1, 6, 52, 602, 8223, 128917, 2273716, 44509914, 957408649, 22449011336, 570032756328, 15587503694363, 456793916757139, 14284890417759141, 474896318288651220, 16726743380843538668, 622282429409944248297, 24385251974172090147514, 1004017088910699487855180
Offset: 0
-
A291843_ser(N, t='t) = {
my(x='x+O('x^N), y=1, y1=0, n=1,
dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
while (n++,
y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
(t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
if (y1 == y, break); y = y1;); y;
};
A291844_ser(N, t='t) = {
my(z = A291843_ser(N+1,t));
((1+x)*z - 1)*(1 + t*x)/((1-t + t*(1+x)*z)*x*z^2);
};
Vec(A291844_ser(20,t=1))
Original entry on oeis.org
36, 994, 22250, 484840, 10867381, 255929070, 6387031115, 169414005231, 4777275646776, 143057565908871, 4541154155493688, 152488745763476252, 5404684397262874216, 201748737356125542014, 7914700977349228045674, 325655486028026921305660, 14026432243484139930962641
Offset: 4
Original entry on oeis.org
16, 1512, 61027, 1977879, 59896915, 1798212190, 55017177704, 1739482570960, 57241033706778, 1967360177317667, 70726834072581710, 2660291453079577988, 104649959742336160214, 4301893149042476310366, 184602995065735203985103, 8259869062587590267542009
Offset: 5
Original entry on oeis.org
1060, 93188, 4823178, 204846125, 8022471066, 305667271592, 11651731789016, 451346936260628, 17927636436397970, 734017001257664973, 31071283511443877004, 1362000990986799549562, 61869212148399394596802, 2912865281703456822415389, 142110906105976971777192795
Offset: 7
Original entry on oeis.org
280, 80632, 7410676, 463514918, 24354193306, 1176805502077, 54779467419573, 2521693624670686, 116608886612461865, 5469515777624179350, 261820716271863054307, 12840622327949419468935, 646781612549311635240576, 33509445393370481956717460, 1787245829075731173749806216
Offset: 8
Showing 1-8 of 8 results.