Displaying 1-10 of 12 results found.
G.f. A(x) satisfies A(x) = (1 + x * A(x)^(5/2)) / (1 - x).
+10
11
1, 2, 7, 32, 167, 942, 5593, 34438, 217888, 1407938, 9252168, 61641846, 415412036, 2826736736, 19395080061, 134034296976, 932110471089, 6518146460274, 45805553781349, 323313555424924, 2291130483593189, 16294149468133930, 116259325138469680
FORMULA
a(n) = Sum_{k=0..n} binomial(n+3*k/2,n-k) * binomial(5*k/2,k) / (3*k/2+1).
G.f. A(x) satisfies:
(1) A(x) = ( 1 + x*A(x)^2/(1 + x*A(x)) )^2.
(2) A(x) = 1/( 1 - x*A(x)^(3/2)/(1 + x*A(x)) )^2.
(3) A(x) = 1 + x * A(x) * (1 + A(x)^(3/2)).
(4) A(x) = B(x)^2 where B(x) is the g.f. of A219537. If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r). (End)
PROG
(PARI) a(n) = sum(k=0, n, binomial(n+3*k/2, n-k)*binomial(5*k/2, k)/(3*k/2+1));
(PARI) a(n, r=2, s=-1, t=4, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 12 2024
G.f. A(x) satisfies A(x) = (1 + x / A(x)^(3/2)) / (1 - x).
+10
9
1, 2, -1, 8, -29, 142, -707, 3714, -20106, 111570, -631046, 3624898, -21089378, 124014048, -735906537, 4401187158, -26501494072, 160532592098, -977574311830, 5981088128586, -36748815585834, 226651808352306, -1402726443269229, 8708648263017666
FORMULA
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(5*k/2-1,k) * binomial(3*k/2-1,n-k) / (5*k/2-1).
PROG
(PARI) a(n) = (-1)^(n-1)*sum(k=0, n, binomial(5*k/2-1, k)*binomial(3*k/2-1, n-k)/(5*k/2-1));
G.f. A(x) satisfies A(x) = (1 + x / A(x)^(5/2)) / (1 - x).
+10
9
1, 2, -3, 22, -138, 1012, -7839, 63506, -531024, 4549276, -39723484, 352237844, -3163252976, 28711196184, -262964888021, 2427319896584, -22557930343459, 210889624536396, -1981972609174109, 18714482015314016, -177453862702083994, 1689045253793239952
FORMULA
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(7*k/2-1,k) * binomial(5*k/2-1,n-k) / (7*k/2-1).
PROG
(PARI) a(n) = (-1)^(n-1)*sum(k=0, n, binomial(7*k/2-1, k)*binomial(5*k/2-1, n-k)/(7*k/2-1));
G.f. A(x) satisfies A(x) = (1 + x * A(x)^(9/2)) / (1 - x).
+10
8
1, 2, 11, 92, 905, 9734, 110867, 1314140, 16041947, 200302394, 2546194497, 32840654064, 428708791851, 5653487876454, 75201937732737, 1007829909427734, 13594917784717860, 184440900147250722, 2515052824018153080, 34451608720123170686, 473853214173320181668
FORMULA
a(n) = Sum_{k=0..n} binomial(n+7*k/2,n-k) * binomial(9*k/2,k) / (7*k/2+1).
PROG
(PARI) a(n) = sum(k=0, n, binomial(n+7*k/2, n-k)*binomial(9*k/2, k)/(7*k/2+1));
G.f. A(x) satisfies A(x) = (1 + x / A(x)^(7/2)) / (1 - x).
+10
8
1, 2, -5, 44, -383, 3782, -39653, 434324, -4910009, 56862170, -671131131, 8043570088, -97629201137, 1197607836678, -14824033357867, 184923041147906, -2322472423266102, 29341825623660226, -372652945642370654, 4755048678561786946, -60929667733382420198
FORMULA
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(9*k/2-1,k) * binomial(7*k/2-1,n-k) / (9*k/2-1).
PROG
(PARI) a(n) = (-1)^(n-1)*sum(k=0, n, binomial(9*k/2-1, k)*binomial(7*k/2-1, n-k)/(9*k/2-1));
G.f. A(x) satisfies A(x) = (1 + x / sqrt(A(x))) / (1 - x).
+10
7
1, 2, 1, 2, 0, 4, -5, 16, -35, 92, -231, 604, -1584, 4214, -11297, 30538, -83096, 227476, -625991, 1730788, -4805594, 13393690, -37458329, 105089230, -295673993, 834086422, -2358641375, 6684761126, -18985057350, 54022715452, -154000562757, 439742222072
FORMULA
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(3*k/2-1,k) * binomial(k/2-1,n-k) / (3*k/2-1).
PROG
(PARI) a(n) = (-1)^(n-1)*sum(k=0, n, binomial(3*k/2-1, k)*binomial(k/2-1, n-k)/(3*k/2-1));
G.f. A(x) satisfies A(x) = (1 + x / A(x)^(9/2)) / (1 - x).
+10
7
1, 2, -7, 74, -820, 10196, -134785, 1860668, -26508457, 386843804, -5753126477, 86878155652, -1328593620692, 20533664196478, -320220157730975, 5032648114664896, -79629405527982623, 1267425784159379572, -20279086501234998596, 325989622456860054852
FORMULA
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(11*k/2-1,k) * binomial(9*k/2-1,n-k) / (11*k/2-1).
PROG
(PARI) a(n) = (-1)^(n-1)*sum(k=0, n, binomial(11*k/2-1, k)*binomial(9*k/2-1, n-k)/(11*k/2-1));
G.f. A(x) satisfies A(x) = 1 + x * (A(x) / (1 - x))^(7/2).
+10
7
1, 1, 7, 49, 378, 3136, 27363, 247597, 2302511, 21872361, 211336755, 2070577285, 20522662832, 205411356794, 2073258075175, 21078157565623, 215658366319375, 2218853063356937, 22942886758494094, 238284942878492146, 2484736162773443446
FORMULA
a(n) = Sum_{k=0..n} binomial(n+5*k/2-1,n-k) * binomial(7*k/2,k) / (5*k/2+1).
PROG
(PARI) a(n) = sum(k=0, n, binomial(n+5*k/2-1, n-k)*binomial(7*k/2, k)/(5*k/2+1));
G.f. A(x) satisfies A(x) = 1 + x + x/A(x)^(5/2).
+10
4
1, 2, -5, 30, -215, 1710, -14516, 128830, -1180920, 11093830, -106245975, 1033454774, -10181848705, 101394979530, -1018972470275, 10320779179380, -105250097458410, 1079767027094630, -11136159773691830, 115395278542757580, -1200814926210284360
FORMULA
G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366401. a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(7*k/2-1,k) * binomial(n+5*k/2-2,n-k) / (7*k/2-1).
PROG
(PARI) a(n) = (-1)^(n-1)*sum(k=0, n, binomial(7*k/2-1, k)*binomial(n+5*k/2-2, n-k)/(7*k/2-1));
CROSSREFS
Cf. A112478, A364393, A364407, A364408, A364409, A366266, A366267, A366268, A366452, A366453, A366454, A366456.
G.f. satisfies A(x) = 1 + x * A(x)^(1/2) * (1 + A(x)^(5/2)).
+10
4
1, 2, 7, 36, 215, 1396, 9571, 68174, 499554, 3741430, 28512825, 220388592, 1723616516, 13614340486, 108450776960, 870264507952, 7028286595932, 57081622558906, 465925734601567, 3820141417134780, 31447663707379395, 259821859662976686, 2153756454578830070
FORMULA
a(n) = Sum{k=0..n} binomial(n,k) * binomial(n/2+5*k/2+1,n)/(n/2+5*k/2+1).
PROG
(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(n/2+5*k/2+1, n)/(n/2+5*k/2+1));
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