A365754
Expansion of (1/x) * Series_Reversion( x*(1-x)/(1+x)^4 ).
Original entry on oeis.org
1, 5, 36, 305, 2833, 27916, 286632, 3033513, 32858595, 362515725, 4059475368, 46021411644, 527163783916, 6092053249160, 70939443268112, 831558454663449, 9804617762941095, 116201796106426543, 1383557994261012100, 16541672701743657545, 198510770031798279825
Offset: 0
-
a(n) = sum(k=0, n, binomial(n+k, k)*binomial(4*(n+1), n-k))/(n+1);
A365753
Expansion of (1/x) * Series_Reversion( x*(1+x)*(1-x)^5 ).
Original entry on oeis.org
1, 4, 27, 220, 1984, 19064, 191325, 1981932, 21031965, 227463808, 2498039219, 27782561352, 312281382836, 3541879743840, 40484779373060, 465888833819532, 5393215780225983, 62761359573224612, 733784067570047400, 8615217370731224160, 101533102164551821896
Offset: 0
-
a(n) = sum(k=0, n, (-1)^k*binomial(n+k, k)*binomial(6*n-k+4, n-k))/(n+1);
A365751
Expansion of (1/x) * Series_Reversion( x*(1+x)*(1-x)^3 ).
Original entry on oeis.org
1, 2, 8, 38, 201, 1134, 6688, 40734, 254237, 1617572, 10452416, 68408626, 452530659, 3020870352, 20324167488, 137672551630, 938154745773, 6426806842566, 44234352581896, 305743015718028, 2121318029754770, 14769052147618740, 103148538125870880
Offset: 0
-
a(n) = sum(k=0, n, (-1)^k*binomial(n+k, k)*binomial(4*n-k+2, n-k))/(n+1);
-
def A365751(n):
h = binomial(4*n + 2, n) * hypergeometric([-n, n + 1], [-4 * n - 2], -1) / (n + 1)
return simplify(h)
print([A365751(n) for n in range(23)]) # Peter Luschny, Sep 20 2023
A365855
Expansion of (1/x) * Series_Reversion( x*(1+x)^2*(1-x)^4 ).
Original entry on oeis.org
1, 2, 9, 46, 264, 1612, 10291, 67830, 458109, 3153744, 22049065, 156127140, 1117369884, 8069610992, 58735003740, 430416574918, 3172987081311, 23514565653058, 175083678670264, 1309132916709168, 9825882638364144, 74003924059921940, 559112987425763365
Offset: 0
-
a(n) = sum(k=0, n, (-1)^k*binomial(2*n+k+1, k)*binomial(5*n-k+3, n-k))/(n+1);
-
def A365855(n):
h = binomial(5*n + 3, n) * hypergeometric([-n, 2*n + 2], [-5 * n - 3], -1) / (n + 1)
return simplify(h)
print([A365855(n) for n in range(23)]) # Peter Luschny, Sep 20 2023
A365878
Expansion of (1/x) * Series_Reversion( x*(1+x)^3*(1-x)^4 ).
Original entry on oeis.org
1, 1, 5, 17, 83, 381, 1939, 9905, 52544, 282315, 1545130, 8552557, 47880020, 270401515, 1539288570, 8821594865, 50860072024, 294774097800, 1716506373521, 10037592274363, 58920231785426, 347051995986538, 2050627029532225, 12151336260368205
Offset: 0
-
a(n) = sum(k=0, n, (-1)^k*binomial(3*n+k+2, k)*binomial(5*n-k+3, n-k))/(n+1);
-
def A365878(n):
h = binomial(5*n + 3, n) * hypergeometric([-n, 3*(n + 1)], [-5 * n - 3], -1) / (n + 1)
return simplify(h)
print([A365878(n) for n in range(24)]) # Peter Luschny, Sep 21 2023
A368079
Expansion of (1/x) * Series_Reversion( x * (1-x)^3 * (1-x^2)^3 ).
Original entry on oeis.org
1, 3, 18, 127, 996, 8322, 72644, 654615, 6043455, 56866028, 543368586, 5258196762, 51426990112, 507537393600, 5048033356128, 50549237164615, 509197913456922, 5156339940802941, 52460340305220466, 535976129228082972, 5496745175387480976
Offset: 0
-
my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3*(1-x^2)^3)/x)
-
a(n, s=2, t=3, u=3) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);
A365765
Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1+x) ).
Original entry on oeis.org
1, 5, 39, 365, 3772, 41491, 476410, 5644477, 68493324, 846937140, 10633195119, 135185288475, 1736883987836, 22516798984946, 294169295918996, 3869084306851933, 51189853304834940, 680816769653570044, 9097058255214149068, 122064057533865334100
Offset: 0
-
CoefficientList[(1/x) *InverseSeries[Series[x*(1-x)^4/(1+x),{x,0,20}]],x] (* Stefano Spezia, May 04 2025 *)
-
a(n) = sum(k=0, n, binomial(n+1, k)*binomial(5*n-k+3, n-k))/(n+1);
A370105
a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k-1,k) * binomial(5*n-k-1,n-k).
Original entry on oeis.org
1, 3, 23, 192, 1687, 15253, 140504, 1311292, 12357015, 117318162, 1120436273, 10752242592, 103596191608, 1001494496863, 9709576926716, 94369011385192, 919175964169623, 8970063281146830, 87685232945278010, 858446087522807784, 8415669293820893937
Offset: 0
-
a(n) = sum(k=0, n, (-1)^k*binomial(n+k-1, k)*binomial(5*n-k-1, n-k));
-
a(n, s=2, t=1, u=3) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((u+1)*n-s*k-1, n-s*k));
Showing 1-8 of 8 results.